Weakly Inscribed Polyhedra

. Motivated by an old question of Steiner, we study convex polyhedra in R P 3 with all their vertices on a sphere, but the polyhedra themselves do not lie on one side the sphere. We give an explicit combinatorial description of the possible combinatorics of such polyhedra. The proof uses a natural extension of the 3-dimensional hyperbolic space by the de Sitter space. Polyhedra with their vertices on the sphere are interpreted as ideal polyhedra in this extended space. We characterize the possible dihedral angles of those ideal polyhedra, as well as the geometric structures induced on their boundaries, which is composed of hyperbolic and de Sitter regions glued along their ideal boundaries.


Introduction
In 1832, Steiner [Ste32, Problem 77] asked the following questions 1 : Does every polyhedron have a combinatorially equivalent realization that is inscribed to a sphere, or to another quadratic surface? If not, which polyhedra have such realizations? In this paper, a polyhedron in R 3 refers to the bounded intersection of finitely many half-spaces or, equivalently, the convex hull of a finitely many points. Given a surface S, a polyhedron P is inscribed to S if all the vertices of P lie on S. We say that a polyhedron is inscribable to S if it has a combinatorially equivalent realization with all its vertices on S.
We use the preposition "to" rather than "in" to make it clear that we do not require that the polyhedron is on one side of S. This is not required in Steiner's definition, either. In fact, since Steiner's problem is obviously projectively invariant, it is quite natural to consider it in projective space. It is then possible that a polyhedron with vertices on the sphere does not lie inside the sphere.
Definition 1.1. In the projective space RP 3 , a polyhedron P inscribed to a quadric S is strongly inscribed to S if the interior of P is disjoint from S, or weakly inscribed to S otherwise.
Remark 1.2. Note in the definition above that weakly inscribed polyhedra exclude those that are strongly inscribed.
Steiner also defined that a polyhedron is circumscribed to a surface if all its facets are tangent to the surface. We will see that circumscription and inscription are closely related through polarity, hence we only need to focus on one of them.
Steiner's problem remained entirely open for nearly a century. There were even beliefs [Brü00] that every simplicial polyhedron is strongly inscribable in a sphere. The first polyhedra without any strongly inscribed realization were discovered in 1928 by Steinitz [Ste28]. It was realized much later that the cube with one vertex truncated cannot be inscribed to any quadric. This follows from the well-known fact that if seven vertices of a cube lie on a quadric, so does the eighth one [BS08, Section 3.2]; see [CP17,Example 4.1] for a complete argument.
There are three quadrics in RP 3 up to projective transformation: the sphere, the one-sheeted hyperboloid, and the cylinder. Strong inscriptions to them are essentially characterized in previous works of Hodgson-Rivin-Smith [HRS92] and Danciger-Maloni-Schlenker [DMS20]. The current paper answers Steiner's question for polyhedra weakly inscribed to a sphere.
The projective space RP 3 can be seen as a completion of the Euclidean space R 3 with a hyperplane at infinity. If the hyperplane at infinity is disjoint from both the sphere and the inscribed polyhedron, then the inscription must be strong (see [CP17]). Hence we focus on the situation where the hyperplane at infinity intersects the sphere, but does not intersect the polyhedron. The sphere then appears in the Euclidean space as a two-sheeted hyperboloid, and the weakly inscribed polyhedron has some vertices on one sheet, and other vertices on the other sheet.
Our main result is the following combinatorial characterization of polyhedra weakly inscribed to a sphere.

Theorem 1.3 (Combinatorial characterization).
A 3-connected planar graph Γ = (V, E) is the 1-skeleton of a polyhedron P ⊂ RP 3 weakly inscribed to a sphere if and only if (C1) Γ admits a vertex-disjoint cycle cover consisting of two cycles and, if we color the edges connecting vertices on the same cycle by red (r), and those connecting vertices from different cycles by blue (b), then 1 The original text in German is 77) Wenn irgend ein convexes Polyëder gegeben ist, lässt sich dann immer (oder in welchen Fällen nur) irgend ein anderes, welches mit ihm in Hinsicht der Art und der Zusammensetzung der Grenzflächenübereinstimmt (oder von gleicher Gattung ist), in oder um eine Kugelfläche, oder in oder um irgend eine andere Fläche zweiten Grades beschreiben (d. h. dass seine Ecken alle in dieser Fläche liegen oder seine Grenzflächen alle diese Fläche berühren)? (C2) there is a cycle visiting all the edges (repetition allowed) along which the edge color has the pattern • . . . bbrbbr. . . if the cycle cover contains a 1-cycle, or • . . . brbr. . . otherwise.
Let us stress again that our definition of weakly inscribed polyhedra excludes those that are strongly inscribed.
Here, we abuse the terminology, and call a single vertex 1-cycle, and a single edge 2-cycle. We will see that the two cycles in (C1) correspond to vertices on the two sheets of the hyperboloid, and the edges are colored blue if they are between the sheets, or red otherwise. If the cycle cover contains a 1-cycle with a single vertex v, Condition (C2) has a much simpler formulation, namely that v is connected to every other vertex. We decide to adopt the current formulation for comparison with the other case. Theorem 1.3 is remarkable because, unlike the characterization of strongly inscribed polyhedra [HRS92], it does not involve any feasibility problem. Despite some efforts [DS96,Che03], no characterization as explicit as Theorem 1.3 has been obtained for strong inscription.
In fact, Theorem 1.3 is a consequence of the following linear programming characterization.
Theorem 1.4 (Linear programming characterization). A 3-connected planar graph Γ = (V, E) is the 1-skeleton of a polyhedron P ⊂ RP 3 weakly inscribed to a sphere if and only if (C1) Γ admits a vertex-disjoint cycle cover consisting of two cycles and, if we color the edges connecting vertices on the same cycle by red, and those connecting vertices from different cycles by blue, there is a weight function w : E → R such that (W1) w > 0 on red edges, and w < 0 on blue edges; (W2) w sums up to 0 over the edges adjacent to a vertex v, except when v is the only vertex in a 1-cycle, in which case w sums up to −2π over the edges adjacent to v.
Recall that we consider a single vertex as a 1-cycle, and a single edge as a 2-cycle. The feasibility problem involved here is visibly much simpler than that in [HRS92]. In particular, there is no inequality on the non-trivial cuts. Recall again that strongly inscribed polyhedra are not weakly inscribed according to our definition.
In [HRS92], the sphere was seen as the ideal boundary of the projective model of the hyperbolic space, so that strongly inscribed polyhedra were interpreted as hyperbolic ideal polyhedra. The characterization is then formulated in terms of hyperbolic exterior dihedral angles. More specifically, a polyhedron is strongly inscribable in a sphere if and only if one can assign weights (angles) to the edges subject to a family of equalities and inequalities. Hence strong inscribability in the sphere can be determined by solving a feasibility problem.
In [DMS20], polyhedra strongly inscribed in a one-sheeted hyperboloid (resp. a cylinder) were seen as ideal polyhedra in the anti-de Sitter space (resp. the half-pipe space [Dan14,Dan13]). Dihedral angles of the ideal polyhedra then lead to a linear programming characterization in the same style of [HRS92].
Remark 1.5. The definition of strong inscription in [DMS20] is slightly stronger than ours. They require that P ∩ S consists of exactly the vertices of P . The two definitions are equivalent only when S is the sphere. Otherwise, it is possible that some edges of P are contained in S.
Our proof to Theorem 1.4 follows a similar approach. Given a sphere S ⊂ RP 3 , its interior (resp. exterior) is seen as the projective model for the 3-dimensional hyperbolic space H 3 (resp. de Sitter space dS 3 ). In [Sch98,Sch01], H 3 and dS 3 together make up the hyperbolic-de Sitter space (HS space for short) which is denoted by HS 3 . Then a polyhedron P inscribed to S can be considered as an ideal polyhedron in HS 3 . We say that P is strongly ideal if the interior of P is contained in H 3 , or weakly ideal otherwise.
We will see in Section 8 that Theorem 1.4 follows from Theorem 3.2 below, which describes the possible dihedral angles of convex polyhedra in HS 3 . More specifically, the dihedral angles at the edges of P form a weight function θ satisfying all the conditions of Theorem 1.4 and, additionally, that |θ| < π and the sum of θ over the blue edges is bigger than −2π. These additional conditions are, however, redundant in the linear programming characterization, as we will prove in Section 8.
Rivin [Riv94] also gave another characterization in terms of the metric induced on the boundaries of ideal hyperbolic polyhedra. More specifically, every complete hyperbolic metric of finite area on an n-times punctured sphere can be isometrically embedded as the boundary of a n-vertices polyhedron strongly inscribed to a sphere, viewed as an ideal hyperbolic polyhedron (possibly degenerate and contained in a plane). Similarly, [DMS20] also characterized polyhedra strongly inscribed in the one-sheeted hyperboloid in termes the possible induced metrics on the boundary of ideal polyhedra in the Anti-de Sitter space.
Extending these previous works, we also provide a characterization for the geometric structure induced on the boundary of a weakly ideal polyhedron in HS 3 . This geometric structure, as distinguished from that induced on a strongly ideal polyhedron, contains a de Sitter part; that is, a part locally modeled on the de Sitter plane. We call this induced data an "HS-structure", since it is locally modeled on HS 2 , a natural extension of the hyperbolic plane by the de Sitter plane. Relevant definitions in the following statement will be recalled in the next section.
Theorem 1.6 (Metric characterization). Let P be a weakly ideal polyhedron in HS 3 with n vertices. Then the induced HS-structure on ∂P is a complete, maximal HS structure on the punctured sphere, obtained by gluing copies of H 2 to a de Sitter surface along their ideal boundaries by C 1 piecewise projective maps such that, at the "break points" where the maps fail to be projective, the second derivative has a positive jump. Conversely, each HS structure of this type is induced on a unique weakly ideal polyhedron in HS 3 .
Note that both the hyperbolic and de Sitter parts of the metric have a well-defined real projective structure at infinity, so it is meaningful to ask for a piecewise projective gluing map. More explanations on the statement of Theorem 1.6 can be found in Section 3.3.
Remark 1.7. For the interest of physics audience, weakly ideal polyhedra can be interpreted as a description of interactions of "photons" in a 3-dimensional spacetime; see [BBS11]. More specifically, HS 2 models the link of an event in a 3-dimensional space-time. The vertices on different boundary components of the de Sitter surface in Theorem 1.6, or, combinatorially, on different cycles in Condition (C1), correspond to incoming and outgoing photons (depending on the direction of time) involved in an interaction. A special case is the single vertex in a degenerate boundary component, or, combinatorially, in a 1-cycle, which corresponds to an extreme BTZ-like singularity.
Remark 1.8. The weak inscription, although covered by Steiner's definition, seemed forgotten and only revived recently. Schulte [Sch87] considered higher dimensional generalizations of Steiner's problem and defined a weaker notion following an idea from [GS87]. However, since he worked in Euclidean space, his definition coincides with the strong inscription. Padrol and the first author [CP17] extended Schulte's definitions into the projective space, and noticed polyhedra inscribed to the sphere but not strongly inscribable.
The paper is organized as follows. The essential definitions are made in Section 2 in a general setting. Then we can view polyhedra weakly inscribed to the sphere as ideal polyhedra in HS 3 . In Section 3, we announce characterizations for the dihedral angles and induced metrics of weakly ideal polyhedra in HS 3 , which are actually reformulations of Theorems 1.4 and 1.6. We also outline the proof strategy, which is carried out in the following sections. In particular, there are some technical challenges which were not encountered in the previous works on strong inscription. For instance, the space of weakly inscribed polyhedra is not simply connected. Finally, in Section 8, we deduce Theorem 1.3 from the linear programming characterization of dihedral angles.

Definitions
We are mainly interested in three dimensional polyhedra. However, the definitions in this section are more general than strictly necessary, and cover the anti-de Sitter and half-pipe spaces that we hope to study in a further work. Lower dimensional cases are used as examples.
2.1. The hyperbolic, anti-de Sitter and half-pipe spaces. The projective space RP d is the set of linear 1-subspaces of R d+1 . An affine chart of RP d is an affine hyperplane H ⊂ R d+1 which is identified to the set of linear 1-dimensional subspaces intersecting H. The linear hyperplane parallel to H is projectivized as the hyperplane at infinity; linear 1-dimensional subspaces contained in this hyperplane have no representation in the affine chart H.
Let R d+1 p,q denotes R d+1 equipped with an inner product ·, · of signature (p, q), p + q ≤ d + 1. We say that R d+1 p,q is non-degenerate if p + q = d + 1. For convenience, we will assume that It is equipped with a complex-valued "distance", which restricts to each connected component as the natural constant curvature metric, and can be defined in terms of the Hilbert metric of the boundary quadric, see [Sch98]. If p and q are both non-zero, HS d p,q consists of a copy of H d p,q and a copy of S d p,q identified along their ideal boundaries. Example 2.2.
• HS d d,1 consists of two copies of the hyperbolic space H d and a copy of the de Sitter space dS d ; we call it the "hyperbolic-de Sitter space", and simplify the notation to HS d .
• Another situation that concerns us in the future is HS d p,p , 2p ≤ d + 1, consisting of two copies of H d p,p differing by the sign of the metric. We denote it by 2H d p,p . In particular, HS 3 2,2 = 2AdS 3 . • Up to a sign of metric, there are five possible HS 2 p,q metrics, namely HS 2 3,0 (= S 2 ), HS 2 2,1 (= HS 2 ), HS 2 2,0 , HS 2 1,1 (= 2HP 2 ) and HS 2 1,0 . • Up to a sign of metric, there are three possible HS 1 p,q metrics. We call a 1-subspace space-, light-or time-like if it is isometric to HS 1 1,1 , HS 1 1,0 or HS 1 2,0 respectively. In an affine chart of RP d p,q , the boundary ∂HS d p,q := ∂H d p,q = ∂S d p,q appears as a quadric in R d . Example 2.3.
• In the affine chart x 3 = 1: H 3 appears as a unit open ball; HP 3 appears as the interior of a circular cylinder; AdS 3 appears as the simply connected side of a one-sheeted hyperboloid. • In the affine chart x 2 = 1: H 3 appears as the two components of the complement of a two-sheeted hyperboloid that do not share a boundary; HP 3 appears as two circular cones; AdS 3 appears as the non-simply connected side of a one-sheeted hyperboloid.
A totally geodesic subspace in HS d p,q is given by a projective subspace L ⊂ RP d . If L is of codimension k, then the induced metric on L is isometric to HS d p ,q for some p + q ≤ d − k and, by Cauchy's interlacing theorem, we have 0 ≤ p−p ≤ k and 0 ≤ q −q ≤ k. If HS d p,q is non-degenerate, then there are three possible metrics on a totally geodesic hyperplane H (codimension 1). We say that H is space-, time-or light-like if it is isometric to HS d−1 p,q−1 , HS d−1 p−1,q or HS d−1 p−1,q−1 , respectively.
The polar of a set X ∈ R d+1 p,q is defined by X * = {x : x, y ≤ 0 for all y ∈ X}.
The polar of a subspace L ⊂ R d+1 p,q is its orthogonal complement, i.e.
If HS d p,q is non-degenerate, L ⊂ RP d is isometric to HS k r,s and L ⊥ to HS k r ,s , then we have If this is the case, X can be identified to a convex cone K in R d+1 that is pointed at the origin and intersects H in a convex set. Then the projectivization of the polar cone K * is the polar of X, denoted again by X * . Equivalently [dGdV58], X ⊂ RP d is convex if for any two points p, q ∈ X, exactly one of the two segments joining p and q is contained in X.
• H d is convex; • AdS d is not convex; • HP d is convex, but its closure is not.
Two convex sets are consistent if some affine chart contains both of them, or inconsistent otherwise.
A convex hull of a set X is a minimal convex set containing X. Note that there is usually more than one convex hull. A convex polyhedron P is a convex hull of finitely many points. A (closed) face of P is the intersection of P with a supporting hyperplane, i.e. a hyperplane that intersects the boundary of P but disjoint from the interior of P . The faces of P decompose the boundary ∂P into a cell complex, giving a face lattice. Two polyhedra are combinatorially equivalent if they have the same face lattice. The polar P * is combinatorially dual to P , i.e. the face lattice of P * is obtained from P by reversing the inclusion relations. We recommend the books [Grü03,Zie95] as general references for polyhedra (or polytopes) theory.
Definition 2.6. A convex polyhedron P ⊂ RP d is ideal to HS d p,q if all its vertices are on the boundary of HS d p,q . An ideal polyhedron P is strongly ideal if the interior of P is disjoint from ∂HS d p,q , or weakly ideal otherwise.
In the case that P is (strongly) ideal to HS d p,q , we also say that P is (strongly) ideal to H d p,q or to S d p,q . A (polyhedral) HS structure of a k-dimensional manifold is a triangulation of the manifold together with an embedding of each k-simplex into RP k p,q , p + q ≤ k + 1, such that the simplices, inheriting the HS k p,q metric, are isometrically identified on their common faces. A convex polyhedron P ⊂ RP d with n vertices naturally induces an HS structure on the n-times punctured S d−1 . If P is ideal to HS d p,q , then this metric is geodesically complete. In any affine chart, ∂HS d p,q appears as a quadratic surface, and an ideal polyhedron appears inscribed to this surface.
For H d , an ideal polyhedron P is strongly ideal if and only if it is consistent with H d [CP17]. Polyhedra strongly ideal to H 3 are then inscribed to a sphere. Their combinatorics was characterized by Hodgson, Rivin and Smith [HRS92]. Polyhedra strongly ideal to HP 3 are inscribed to a circular cylinder. Polyhedra strongly ideal to AdS 3 are inscribed to and contained in a onesheeted hyperboloid. Danciger, Maloni and the second author [DMS20] have essentially provided characterizations of the combinatoric types of these polyhedra.
We will focus on weakly ideal polyhedra, i.e. ideal polyhedra that are not strongly ideal. We prefer affine charts that contains the polyhedron P ; such an affine charts cannot contain H 3 , HP 3 or AdS 3 by the discussion above. Polyhedra weakly ideal to H 3 are then inscribed to a two-sheeted hyperboloid. Polyhedra weakly ideal to HP 3 are inscribed to a circular cone. And finally, polyhedra weakly ideal to AdS 3 are inscribed to, but not contained in, a one-sheeted hyperboloid. This covers all the quadratic surfaces, and characterizing the weakly ideal polyhedra in H 3 , HP 3 and AdS 3 would provides a complete answer to Steiner's problem.

Overview
From now on, we will focus on projective polyhedra weakly inscribed to the sphere, which is equivalent to projective polyhedra weakly ideal to H 3 , or Euclidean polyhedra inscribed to the two-sheeted hyperboloid.
Recall that a polyhedron P weakly ideal to H 3 is not consistent with H 3 . Since we prefer affine charts containing P , H 3 would appear, up to a projective transformation, as the set x 2 0 +x 2 1 −x 2 2 < −1 in such charts. This is projectively equivalent to the Klein model. We use H 3 + and H 3 − to denote the parts of H 3 with x 2 > 0 and x 2 < 0, respectively. Moreover, the boundary ∂H 3 appears as a two-sheeted hyperboloid.
3.1. Ideal polyhedra. For a polyhedron P weakly ideal to H 3 , let V denotes the set of its vertices; then V ⊂ ∂H 3 by definition. We write V + = V ∩ ∂H 3 + and V − = V ∩ ∂H 3 − , and say that P is (p, q)-ideal if |V + | = p and |V − | = q. P is strongly ideal if p = 0 or q = 0; we only consider weakly ideal polyhedra, hence p > 0 and q > 0. If p, q > 1, we orient the curves P ∩ ∂H 3 ± so that they are homologous on ∂P . We then label the vertices of V + by 1 + , . . . , p + , and vertices of V − by 1 − , . . . , q − , in the order compatible with the orientation.
Let P n denote the space of labeled polyhedra with n ≥ 4 vertices that are weakly ideal to H 3 , considered up to hyperbolic isometries, and P p,q denote the space of labeled (p, q)-ideal polyhedra, p + q ≥ 4. Then P n is the disjoint union of P p,q with p + q = n. We only need to study connected components P p,q , and may assume p ≤ q without loss of generality. We usually distinguish two cases, namely p < 2 < q and 2 ≤ p ≤ q. Note that we always assume that p ≥ 1 since we only consider only weakly ideal polyhedra, so p < 2 below always means p = 1.
3.2. Admissible graphs. We define a weighted graph (or simply graph) on a set of vertices V as a real valued function w defined on the unordered pairs |V | 2 . The weight w v at a vertex v ∈ V is defined as the sum u w(u, v) over all u = v.
Unless stated otherwise, the support of w is understood as the set E of edges. We can treat w as a usual graph with edge weights, and talk about notions such as subgraph, planarity and connectedness. But we will also take the liberty to include edges of zero weight, as long as it does not destroy the property in the center of our interest. For example, graphs in this paper are used to describe the 1-skeleta of polyhedra, i.e. 3-connected planar graphs. Hence whenever convenient, we will consider maximal planar triangulations. If this is not the case with the support of w, we just triangulate the non-triangle faces by including edges of zero weight.
The advantage of this unconventional definition is that graphs can be treated as vectors in . Weighted graphs of a fixed combinatorics, together with their subgraphs, then form a linear subspace. Graphs with a common subgraph correspond to subspaces with nontrivial intersection. This makes it convenient to talk about neighborhood, convergence, etc. For a fixed polyhedral combinatorics, our main result implies that the set of weighted graphs form a (|E|−|V |)-dimensional cell. Weighted graphs of a fixed number of vertices then form a cell complex of dimension 2|V | − 6 in R ( |V | 2 ) : The maximal cells correspond to triangulated (maximal) planar graphs, and they are glued along their faces corresponding to common subgraphs.
Consider an edge e of an ideal polyhedron P . Then e is either a geodesic in H 3 , or a time-like geodesic in dS 3 . In both cases, the faces bounded by e expand to half-planes forming a hyperbolic exterior dihedral angle, denoted by ϑ. We assign to e the HS exterior dihedral angle θ, which equals ϑ if e ⊂ H 3 , or −ϑ if e ⊂ dS 3 . We will refer to θ as exterior angles, dihedral angles, or simply angles, and should not cause any confusion.
This angle assignment induces a graph on V , also denoted by θ, supported by the edges of P . We have thus obtained a function Θ that maps an ideal polyhedron P to the graph θ of its angles. Obviously, Θ(P ) is polyhedral, i.e. 3-connected planar. We will see that, if P is (p, q)-ideal, then (C1) θ = Θ(P ) admits a vertex-disjoint cycle cover consisting of a p-cycle and a q-cycle and, if we color the edges connecting vertices on the same cycle by red, and those connecting vertices from different cycles by blue, then (A1) 0 < θ < π on red edges, and −π < θ < 0 on blue edges; The sum of θ over blue edges is ≤ −2π, and the equality only happens when p < 2 < q.
Given a (p, q)-admissible graph drawn on the plane, we may orient the p-cycle and the q-cycle so that they are homologous in the 1-point compactified plane. Then we label the vertices on the p-cycle by 1 + , . . . , p + , and vertices on the q-cycle by 1 − , . . . , q − , according to this orientation. Let A p,q denote the space of labeled (p, q)-admissible graphs with p + q ≥ 4. We use A n , n ≥ 4, to denote the disjoint union of A p,q with p + q = n. Our main Theorem 1.4 is the consequence of the following theorem: Theorem 3.2. Θ is a homeomorphism from P p,q to A p,q .
Note that Condition (A3) and some inequalities in Condition (A1) are not present in Theorem 1.4. We will see that they are indeed redundant when formulating a feasibility problem.
3.3. Admissible HS structures. Let ∆ denote the function that maps an ideal polyhedron to its induced HS structure. If P is (p, q)-ideal, it follows from the definition that ∆(P ) is geodesically complete, and it is maximal in the sense that it does not embed isometrically as a proper subset of another HS structure.
The part of ∂P in H 3 has no interior vertex, hence is isometric to a disjoint union of copies of H 2 . In the case 2 ≤ p ≤ q, we use H 2 ± to denote the copies induced by ∂P ∩ H 3 ± . If p < 2 < q, we have only H 2 − = ∂P ∩ H 3 − . The part of ∂P in dS 3 has no interior vertex, neither, hence ∂P ∩ dS 3 is isometric to a complete de Sitter surface.
The intersection of ∂P with a space-like plane in dS 3 is a simple polygonal closed space-like curve in ∂P ∩ dS 3 . If 2 ≤ p ≤ q, this polygonal curve can be deformed to one of maximal length, say γ 0 , which is therefore geodesic in ∂P ∩ dS 3 . Considered as a polygonal curve in dS 3 , γ 0 is then E-convex in the sense of [Sch98, Def 7.13], and it follows that its length is less than 2π, see [Sch98, Prop 7.14]. As a consequence, γ 0 is the unique simple closed space-like geodesic in ∂P ∩ dS 3 , because any other simple closed space-like geodesic would need to cross γ 0 at least twice (there is no de Sitter annulus with space-like, geodesic boundary by the Gauss-Bonnet formula), and two successive intersection points would be separated by a distance π, leading to a contradiction. We denote the metric space ∂P ∩ dS 3 by dS 2 . dS 2 has two boundary components, both homeomorphic to a circle.
If p < 2 < q, then one boundary component of S degenerates to a point. In this case, the metric space ∂P ∩ dS 3 does not contain any closed space-like geodesic, and we denote it by dS 2 0 . Hence ∆(P ) is obtained by gluing one or two copies of H 2 to the non-degenerate boundary components of a de Sitter surface. Let γ ± be the map that glues ∂H 2 ± to ∂dS 2 . We will see that γ ± are C 1 piecewise projective maps (CPP maps for short). More specifically, they are projective except at the vertices of P . The points where the map is not projective are called break points. A break point is said to be positive (resp. negative) if the jump in the second derivative at this point is positive (resp. negative). We will see that the break points of γ ± are all positive.
Definition 3.3. A (p, q)-admissible HS structure, p + q ≥ 4, is obtained In the case p < 2 < q: by gluing a copy of H 2 to dS 2 0 along the non-degenerate ideal boundary by a CPP map with q positive break points. In the case 2 ≤ p ≤ q: by gluing two copies of H 2 to dS 2 , 0 < < 2π, along the ideal boundaries through CPP maps with, respectively, p and q positive break points.
In Figure 1 we sketch the situation of p = 2 and q = 3 is sketched in Figure 1. Figure 1. A sketch of a (2, 3)-admissible HS structure. The dashed segments indicates the gluing maps, including one that produces dS 2 , and two CPP maps with two and three break points, respectively.
Given a (p, q)-admissible HS structure, we may label the break points in the two boundary components of S by 1 + , . . . , p + and 1 − , . . . , q − , respectively. Let M p,q denote the space of (p, q)ideal HS structures up to isometries. Our main Theorem 1.6 is the consequence of the following theorem.
Theorem 3.4. ∆ is a homeomorphism from P p,q to M p,q .
3.4. Outline of proofs. We will prove that Θ and ∆ are local immersions (Section 6) with images in A p,q (Section 4.2) and M p,q (Section 4.3), respectively. They are then local homeomorphisms because P p,q , A p,q and M p,q have the same dimension 2(p + q − 3) (Section 7). Moreover, they are proper maps (Section 5), hence are covering maps. A difference from the previous works lies in the fact that P p,q , A p,q and M p,q are not simply connected if 2 ≤ p ≤ q. We will use open covers and universal covers to conclude that the covering numbers of Θ and ∆ are one (Section 7). 4. Necessity 4.1. Combinatorial conditions. We first verify combinatorial Condition (C1). For this we will need some lemmata about convex sets in RP d . Proof. Consider an affine chart containing A. In this chart, B is either connected, or composed of two connected components which are both convex. Each of those connected components has an intersection with A which is either empty or convex. Therefore, the intersection A ∩ B has at most two connected components.
If A and B are consistent, we can regard them as convex sets in Euclidean space. Hence they are either disjoint, or their intersection is convex, hence connected.
Suppose now that A is disjoint from B. They can then be lifted to disjoint convex subsets A and B in S d . By the spherical hyperplane separation theorem, there exists a spherical hyperplane H ⊂ S d disjoint from A and B . The projection H from H to RP d is then disjoint from A and B, and its complement is then an affine chart containing A and B. This shows that A and B are consistent.
Finally suppose that A and B have a connected intersection, which we call C. Since A and B are convex, C is then convex. We can then lift A, B and C to subsets A , B and C of S d , in such a way that A ∩ B = C . We claim that A ∩ (−B ) = ∅. Indeed, let x ∈ C = A ∩ B , and suppose that there is another point y ∈ A ∩ (−B ). Let x and y be the projections of x and y on RP d , so that x, y ∈ C. Since C is convex, there is a segment, which we denote by [x, y], which connects x to y in C. We can lift this segment to C , so that y lifts to an inverse image y ∈ {y , −y } in C . Since y ∈ A , −y ∈ A , and therefore −y ∈ C , and thus y = y . It follows that y ∈ B , a contradiction. So A ∩ (−B ) = ∅, as claimed.
We can now apply the spherical hyperplane separation theorem: there is a hyperplane H ⊂ S d which separates A from −B . The projection H of H to RP d is then disjoint from A and from B, and the corresponding affine chart contains both A and B.
It follows that if A ∩ B is connected, then A and B are consistent. Therefore, A and B are consistent if and only if A ∩ B is empty or connected.
If A and B are two inconsistent convex regions in RP 2 , then ∂A ∩ ∂B consists of at most four connected components, at most two on the boundary of each connected component of A ∩ B. Otherwise, either the interior or the closure of A ∩ B would consist of more than two connected components, contradicting the lemma above.
A particular case is when A = H 2 and B is a polygon, and their boundaries intersect at the vertices of B, i.e. B is weakly ideal to HS 2 . In this case, Lemma 4.1 implies Corollary 4.2.
• Any convex polygon strongly ideal to HS 2 with at least three vertices is disjoint from dS 2 .
• Any convex polygon ideal to HS 2 with at least five vertices is strongly ideal in HS 2 .
• A weakly ideal convex polygon P has three or four vertices, at most two in each connected component of ∂P ∩ H 2 .
A dual version of this corollary was proved in [CP17]. Notice that there is only one possibility for a weakly ideal triangle; see Figure 2. Moreover, it is known that every connected component of A ∩ B is convex [Tod10]. Proof. We only need to argue for ∂C 2 ∩ ∂B. The other cases follow similarly.
We work in an affine chart containing A; thus it does not contain B. Let p ∈ ∂C 1 ∩ ∂B. Then for any q ∈ ∂C 2 ∩ ∂B, the bounded closed segment [pq] is disjoint from the interior of B, hence also from the interior of C 2 . On the other hand, any q ∈ ∂C 2 \ ∂B (if not empty) is in the interior of B, hence [pq ] must intersect the interior of C 2 . In other words, ∂C 2 ∩ ∂B is the part of ∂C 2 "visible" from p, which must be contractible as C 2 is.
The proof for d = 2 is illustrated in Figure 3.  Proof. A triangular facet F that is not interior would be a weakly ideal in span(F ). Hence P has no exterior facet. The only exterior faces are edges. Then we observe from Figure 2 that, whenever F is not interior, F ∩ ∂H 3 is an arc which is isotopic to the unique interior edge of F . This isotopy induces a homotopy from ∂P ∩ H 3 to I(P ). The former is contractible by Lemma 4.3, hence so is I(P ).
The vertices of P are all on the boundaries of ∂P ∩ H 3 (and of I(P )). Otherwise, through a vertex in the interior of ∂P ∩ H 3 , we can find an hyperplane H such that the intersection of H ∩ ∂P and H ∩ ∂H 3 consists of more than two connected components, contracting Lemma 4.1. Hence the vertices of each component induce an outerplanar graph. If there are at least three vertices in this component, the boundary edges form a Hamiltonian cycle (of the induced graph), so the outerplanar graph is 2-connected. We then conclude that the component of I(P ) is homeomorphic to a disk.
This proves the necessity of Condition (C1) since the boundary edges of the 2-connected outerplanar graphs form a cycle cover consisting of two cycles.
Induction in higher dimensions yields that the union of interior faces consists of two contractible components, and vertices and edges in each component form a 2-connected graph. However, it is possible that a component is not homeomorphic to the (d − 1)-ball, as shown in the following example.
If is sufficiently small, the polyhedron has two interior facets in H 4 + , whose intersection is a single edge connecting [ √ 2, ±1, 0, 0]. Hence the union of interior faces is not homeomorphic to a 3-ball.
4.2. Angle conditions. We now prove that the conditions in Definition 3.1 are necessarily satisfied by the graph induced by the angles of a (p, q)-ideal polyhedron..
For that, we need to verify all the conditions defining an admissible graph. We color interior edges by red, and exterior edges by blue, and prove the angle conditions with respect to this coloring. In the previous part we have seen that this coloring coincides with the combinatorial description in Theorem 3.2.
Among the conditions that involve angles, Condition (A1) comes from the definition of angle. To see Condition (A2) we need the vertex figures.
Recall that a horosphere in H d based at an ideal point x ∈ ∂H d is a hypersurface that intersects orthogonally all the geodesics emerging from x. In the projective model of H 3 , horospheres appear as flattened spheres tangent to the hyperbolic boundary. Similarly, a horosphere in dS d based at x ∈ ∂dS d is a hypersurface that intersects orthogonally all the geodesics emerging from x.  Now consider an ideal polyhedron P . The vertex figure of P at a vertex v, denoted by P/v, is the projection of P with respect to v. P/v is therefore a polygon in RP 2 . A chart is provided by a horosphere in HS 3 based at v. This chart sends v ⊥ to the line at infinity, hence does not contain P/v unless the neighborhood of v in ∂P is contained in H 3 or in dS 3 .   Figure 5) that he turns anti-clockwise at the vertices corresponding to red edges, and clockwise at the vertices corresponding to blue edges. Indeed, the turning direction switches when the walker passes through infinity. The turning angles are then the dihedral angles θ at the corresponding edges, taking anti-clockwise turns as positive, and clockwise turns as negative. Condition (A2) then follows immediately.
We now verify the last angle condition.
Proposition 4.7 (Condition (A3)). If P is weakly ideal, then Θ(P ) sum up to at least −2π over the blue edges, and −2π is achieved only when p < 2 < q.
Proof. It is clear from the vertex figure that the sum over blue edges is −2π when p = 1. Hence we focus on the case 2 ≤ p ≤ q and prove that the sum over blue edges is strictly larger than −2π.
If 2 ≤ p ≤ q, the polar P * of P is a compact polyhedron in dS 3 . All its faces are light-like (isotropic). The edges of P * polar to the blue edges of P form a closed space-like polygonal curve γ, whose vertices are polar to the non-interior facets of P .
The polygonal curve γ is a T -geodesic for the induced HS-structure on P * ; see [Sch01, Definition 3.4]. It then follows from point C. in [Sch01, Theorem 1.5] that γ has length strictly less than 2π. The proposition follows by polarity.
4.3. Metric conditions. We show in this part that the conditions of Theorem 1.6 are necessary.
If P is (p, q)-ideal, we have argued that ∆(P ) consists of one or two copies of H 2 and a de Sitter surface. It remains to verify that the pieces are glued along their ideal boundaries by CPP (C 1 piecewise projective) maps with positive break points at the vertices of P . We may focus on the boundary component of dS 2 , ≥ 0, consisting of q > 1 vertices of P . Let γ be the map that sends ∂H 2 to this boundary.
We can identify ∂H 2 to the real projective line RP 1 . Let v 0 < v 1 < · · · < v p = v 0 , in this order, be the p vertices of P . They divide RP 1 into p segments [v i , v i+1 ]. Each segment corresponds to a segment of ∂HS 2 in the interior of a face triangle weakly ideal to HS 2 . Hence for each i, 0 ≤ i < p, the restriction γ i = γ| [vi,vi+1] is projective. This proves that γ is piecewise projective.
To study differentiabilities, we will follow the work of Martin [Mar05] on CPP homeomorphisms of RP 1 . Note that ∂dS 2 is not projectively equivalent to RP 1 . But our CPP maps are local homeomorphisms. Hence the definitions and many results from [Mar05] remain valid for our case. . For x ∈ RP 1 , let γ ← x ∈ PSL(2, R) (resp. γ → x ∈ PSL(2, R)) be the left (resp. right) germ of a piece-wise projective map γ. The projective transformation D x γ = (γ → x ) −1 • γ ← x is called the shift of γ at x. Note that in this definition, γ ← x and γ → x depend on a local identification of the target with RP 1 , but D x , γ → x and D x γ extend uniquely to isometries of HS 2 . We abuse the same notations for these extensions. The shift D x γ measures how much γ fails to be projective at x. It can be understood as the holonomy of the HS structure along a curve going around x. The following lemma reveals the relation between the shift and the differentiability of γ. Proof. The proof of [Mar05, Proposition 2.3] can be used here, word by word, to prove the "only if". For the "if" part, assume x = 0. Note that u := D 0 γ : RP 1 → RP 1 fixes 0. If (the extension of) u preserves the horocycles based at 0, it must be of the form u(t) = t/(ct + 1). Hence u (0) = 1, therefore γ is C 1 .
Then γ being C 1 follows from the following proposition.
Proposition 4.11. For any x ∈ ∂H 2 , D x γ preserves the horocycles based at x.
Proof. As a measure of how much γ fails to be projective at x, D x γ must be trivial except at vertices of of P . Let v ∈ V + be a vertex of P .
We first note that in a neighborhood of v, exactly two faces of ∂P , say F ← and F → , have non-empty intersections with ∂H 3 . This can be seen from the vertex figure P/v. Recall that vertices (resp. edges) of P/v correspond to edges (resp. faces) of P adjacent to v. In the affine chart of RP 2 provided by a horosphere at v, the tangent plane of ∂H 3 at v is sent to infinity, and intersects P/v in exactly two edges, corresponding to F ← and F → . In particular, no vertex of P/v is at infinity; otherwise such a vertex would correspond to an edge e of P tangent to the quartic ∂H 3 , and the other end of e can not lie on the same quartic.
Our gluing map γ coincides with γ ← v on ∂HS 3 ∩ F ← , and with γ → v on ∂HS 3 ∩ F → . F ← and F → extend to half-planes bounding a dihedral angle Φ containing P . The boundary of Φ is isometric to HS 2 . Leth be a horocycle in HS 3 based at v. Then the intersection h = ∂Φ ∩h give a horocycle in HS 2 . On the other hand, h = ∂P ∩h give a horocycle in ∂P based at v. In a neighborhood of v, h lies within F ← and F → .
We then conclude that, for any horocycle h in Proposition 4.11 can be interpreted as horocycles "closing up" after going around a vertex. In Rivin's characterization of polyhedra strongly inscribed in the sphere, the same phenomenon is reflected by the shearing coordinates summing up to 0 around each vertex. We can do the same with a proper definition of shearing coordinates.
Note that three vertices on ∂HS 2 determine a unique strongly ideal (hyperbolic) triangle -in other words, any three distinct points on the boundary circle of HS 2 determine a unique triangle contained in the disk, corresponding here to the hyperbolic plane. Hence for a given ideal HS structure δ ∈ M p,q , we can replace every triangle in δ, if not already strongly ideal, by the unique hyperbolic triangle with the same vertices. The result is a hyperbolic structure η (a triangulation with hyperbolic simplices) of the n-times punctured S 2 (not embedded). We define the HS shearing (or simply shearing) along an edge of δ as the hyperbolic shearing (see [Pen87,Pen12]) along the corresponding edge of η.
Shearing can be easily read from the vertex figures. First note that the vertex figure of η at a vertex v can be obtained from that of δ by replacing the segments through infinity by the unique other segments with the same vertices. For example, Figure 6 is obtained from Figure 5. Let e be an edge of P adjacent to a vertex v. The shearing along e then equals to the logarithm of the length ratio of the segments adjacent to e in the vertex figure of η at v. Horocycles in η close up if and only if the hyperbolic shearings sum up to 0 over the edges adjacent to v. Then we see from the vertex figure that the horocycles in δ also "close up". And by definition, the HS shearings of η must also sum up to 0. Different triangulation of δ would yield a different hyperbolic metric η. But for a fixed triangulation, it is well-known (see [Pen87,Pen12]) that the hyperbolic shearing on the edges of η provide a coordinate system for the hyperbolic structure. Hence the HS shearing on the edges of δ provide a coordinate system for the ideal HS metrics. Now back to the proof of necessity. Proposition 4.11 asserts that the D x γ are parabolic transformations for every x ∈ RP 1 . Consider the projective transformation u x : t → 1/(t − x) sending x to infinity. Then the conjugate u x D x (γ)u −1 x is a translation of the form t → t + d x (γ). We have d x (γ) = 0 at projective points. At break points: Lemma 4.12. d x (γ) < 0 (resp. > 0) if x is a positive (resp. negative) break points.
Proof. We may assume x = 0, then u 0 = 1/t. We already figured that D 0 (γ) is of the form D 0 (γ) : t → t/(ct + 1). So the conjugate u 0 D 0 (γ)u −1 0 has the form t → t + c, i.e. d 0 (γ) = c. On the other hand, an elementary computation shows that the second derivative d 2 dt 2 | t=0 D 0 (γ) = −2c. In the half-space model of H 2 , let h be a horocycle based at x. Then d x (γ) < 0 (resp. > 0) if and only if D x γ moves points on h in the clockwise (resp. anti-clockwise) direction. We are now ready to prove that every break point of γ is positive.  Remark 4.14. Lemma 6.5 of [Mar05] asserts that d x (γ) equals the change of length of a well chosen segment of horocycle based at x. Up to a scaling, this also suffices for us to conclude that d v (γ) < 0.

Properness
The following theorem states that the maps Θ and ∆ are proper.
Theorem 5.1. Consider a sequence of polyhedra (P k ) k∈N that exits every compact in P p,q , then θ k = Θ(P k ) exits every compact in A p,q and δ k = ∆(P k ) exits every compact in M p,q .
Up to hyperbolic isometries, we may fix three vertices for every polyhedron in (P k ). As ∂H 3 is compact, we may assume that vertices of P k have well defined limits by taking a subsequence. But the limit of P k , denoted by P ∞ , is not a (p, q)-ideal polyhedron, since the sequence exits every compact.
Hence in the limit, P ∞ must fail to be strictly convex at some vertex v. Let P ∞ be the convex hull of all the other vertices. There are three possibilities, namely that v is in the relative interior of a vertex, an edge or a facet of P ∞ . But every straight line intersect a quadratic surface in at most two points, hence an ideal vertex can not be in the relative interior of an edge. Thus we only need to consider the remaining two possibilities.
Remark 5.2. For strongly ideal polyhedra, an ideal vertex can not lie in the interior of a facet. Hence there is only one possibility to consider. See [DMS20].
Proposition 5.3. If some vertices of (P k ) k∈N converge to the same vertex of P ∞ , then the limit graph θ ∞ violates Condition (A2) or (A3), and some break points in δ k merge into a single break point in the limit metric δ ∞ .
Proof. The divergence of the induced metrics follows immediately from the correspondence between vertices of P and break points in ∆(P ). Hence we will focus on the divergence of the admissible graphs.
Note that the ideal boundary of H 3 can be seen as two copies of H 2 identified along their ideal boundaries. With our choice of affine chart for the projective model, the two copies of H 2 appear as the two sheets of the hyperboloid x 2 0 + x 2 1 − x 2 2 = −1. The vertex set of each P k ∈ P p,q then corresponds to two point sets V + k and V − k in H 2 of cardinality p and q respectively. They converge to two point sets V + ∞ and V − ∞ of cardinality p and q respectively, corresponding to the vertices of P ∞ . If some vertices of (P k ) converges to the same vertex of P ∞ , we must have p + q < p + q. Up to hyperbolic isometries, we may assume three fixed vertices shared by all P k , hence 3 ≤ p + q .
Consequently, the graph Θ(P ∞ ) has at least three vertices, but stricly less vertices than θ ∞ . In fact, it is obtained by contracting vertices of θ ∞ . Recall that θ k = Θ(P k ) and their limit θ ∞ are vectors of dimension |V | 2 = p+q 2 . Assume that a set of vertices S ⊂ V is merged into a vertex of P ∞ . If S = V + or V − , we have on the one hand On the other hand, as the limit of θ k , Condition (A2) asserts that for any v ∈ S. Comparing the two sums, we conclude that Now assume that θ ∞ is (p, q)-admissible. Since S are vertices on the same polar circle, θ ∞ (u, v) is non-negative, hence must be 0, if u, v ∈ S. In other words, S induce an empty graph, contradicting the fact that vertices in S are consecutive in a cycle. If S = V + or V − , we must have p > 1. But it is easy to conclude that the sum over negative weights in θ ∞ is −2π, contradicting Condition (A3).
Proposition 5.4. If a vertex v of P k converges to a vertex of P ∞ that is contained in a unique supporting plane, then v is an isolated vertex in the limit graph θ ∞ , and v is not a break point in the HS structure induced by δ ∞ .
Proof. Under the assumption of the proposition, every face of P ∞ adjacent to v must lie in this unique supporting plane. Otherwise, the supporting plane of the face would be another supporting plane containing v. Then the dihedral angles vanish on all the edges incident to v. Since v is in the interior of a weakly ideal HS triangle, the gluing map is projective at v.
A special case is of particular importance for us: If ∂P k converges to a double cover of a plane, then the limit polyhedron P ∞ is equal to RP 3 . In this case, every vertex is "flat": θ ∞ is identically 0 (empty graph), and δ ∞ is the double cover of HS 2 . We call this polyhedron a flat polyhedron.
Proposition 5.5. If two faces of P ∞ intersect in their relative interiors and span a plane, then P ∞ is flat.
Proof. Under the assumption of the proposition, the only plane that avoids the interior of P ∞ is the plane spanned by the two intersecting faces. Hence every face admits this plane as the supporting plane. In other words, every face lies in this plane. 6. Rigidity 6.1. The infinitesimal Pogorelov map. We recall here the definition of the infinitesimal Pogorelov map, as well as its key properties. We refer to [Sch98] for the proofs, see in particular Définition 5.6 and Proposition 5.7 in [Sch98]. Other relevant references are [Fil07,Izm09,LS00,Sch05].
With affine charts containing weakly ideal polyhedra, the hyperplane at infinity H ∞ is space-like. Apart from the HS metric and the usual Euclidean metric, the affine charts can also carry the Minkowski metric. Then the point x 0 = H ⊥ ∞ is the "center" of the Minkowski space R 2,1 . The set of light-like geodesics passing through x 0 is called the light cone at x 0 , denoted by C(x 0 ).
Let U = RP 3 \ H ∞ be an affine chart, and ι : U → R 2,1 be the projective embedding into the Minkowski space. The infinitesimal Pogorelov map Υ is then defined as the bundle map Υ : T U → T R 2,1 over the inclusion ι : U → R 2,1 as follows: Υ agrees with dι on T x0 U . For any x ∈ U \ C(x 0 ), and any vector v ∈ T x U , write v = v r + v ⊥ , where v r is tangent to the radial geodesic passing through x 0 and x, and v ⊥ is orthogonal to this radial geodesic, and define where the norm · HS in the numerator of the first term is the HS metric, the norm · 2,1 in the denominator is the Minkowski metric andx is the normalized radial vector (so x 2 2,1 = ±1). The key property of the infinitesimal Pogorelov map is the following (the proof is an easy computation in coordinates, that can be adapted from [Fil11, Lemma 3.4]).
Lemma 6.1. Let Z be a vector field on U \ C(x 0 ) ⊂ HS 3 . Then Z is a Killing vector field if and only if Υ(Z) (wherever defined) is a Killing vector field for the Minkowski metric on R 2,1 .
In fact, the lemma implies that the bundle map Υ, which so far has only been defined over U \ C(x 0 ), has a continuous extension to all of U . The bundle map Υ is called an infinitesimal Pogorelov map.
Next, the bundle map Ξ : T R 2,1 → T R 3 over the identity, which simply changes the sign of the n-th coordinate of a given tangent vector, has the same property: it sends Killing vector fields in R 2,1 to Killing vector fields for the Euclidean metric on R 3 . Hence the map Π = Ξ • Υ is a bundle map over the inclusion U → R 3 with the following property: Lemma 6.2. Let Z be a vector field on U ⊂ HS 3 . Then Z is a Killing vector field if and only if Π(Z) is a Killing vector field for the Euclidean metric on R 3 .
The bundle map Π is also called an infinitesimal Pogorelov map, since it is an infinitesimal version of a remarkable map introduced by Pogorelov [Pog73] to handle rigidity questions in spaces of constant curvature. 6.2. Rigidity with respect to HS structures. Here, an infinitesimal deformation of P associates a vector tangent to ∂H 3 to each vertex of P ; the infinitesimal deformation is trivial if it is the restriction of a global Killing field of HS 3 . Proposition 6.3. Let P ∈ P p,q andṖ be an infinitesimal deformation of P within P p,q . IfṖ does not change the HS structure ∆(P ) at first order, thenṖ is trivial.
Proof. As always, we work in an affine chart containing P . Suppose thatṖ is a non-trivial infinitesimal deformation of P that does not change, at first order, the HS metric induced on P . Then the induced HS metric on each facet is constant at first order. Hence for each facet F , there is a Killing field κ F such that the restriction of κ F to the vertices of F is equal to the restriction oḟ P , and for two facets F and G, κ F and κ G agree on the common edge of F and G.
Lemma 6.2 shows thatκ F = Π(κ F ) is the restriction of a Killing field of R 3 , while it is clear that if F and G share an edge, thenκ F andκ G agree on this edge. Therefore the restriction ofκ F to the vertices of P define an isometric first-order Euclidean deformation of P .
However, Alexandrov [Ale05] proved that convex polyhedra in R 3 are infinitesimally rigid: any first-order Euclidean isometric deformation must be the restriction of a global Killing vector field of R 3 . Soκ F must be the restriction of a global Killing vector fieldκ. Lemma 6.2 therefore implies that κ F are the restriction to the faces of P of a global Killing vector field κ = Π −1 (κ), which contradicts our hypothesis. 6.3. Shape parameters associated to edges. We can identify ∂H 3 with the extended complex plane CP 1 , then vertices of an ideal polyhedron P can be described by complex numbers. By subdividing non-triangular facets if necessary, we may assume that facets of P are all triangles. Let z 1 z 2 z 3 and z 2 z 1 z 4 be two facets of P oriented with outward pointing normal vectors. The shape parameter on the common (oriented) edge z 1 z 2 is the cross ratio Recall that each triangular facet of P determines a strongly ideal triangles with the same vertices. The two (oriented) hyperbolic triangles corresponding to z 1 z 2 z 3 and z 2 z 1 z 4 form a hyperbolic dihedral angle at their common edge z 1 z 2 . Let φ denote the hyperbolic exterior angle at z 1 z 2 . Then the shape parameter τ has a geometric interpretation: it can be written in the form of τ = exp(σ + iφ), where σ is the shearing between the two hyperbolic triangles.
The angle φ can be read from the hyperbolic vertex figure (see Section 4.3). If one walks along the polygonal curve, in the same direction as we specified for reading θ (see Section 3), then φ is nothing but the turning angle at every vertex, taking anti-clockwise turns as positive, and clockwise turns as negative.
Let v be a vertex of P , and let τ 1 , τ 2 , · · · , τ k be the shape parameters associated to the edges of P adjacent to v, in this cyclic order. The following relations, which holds for strongly inscribed polyhedra, also holds for the weakly inscribed P .
Both equations can be easily understood by considering the hyperbolic vertex figure at v: (1) follows from the fact that σ i = 0 while φ i is a multiple of 2π.
(2) is just saying that the vertex figure, considered as a polygonal curve in the Euclidean plane, closes up.
The shape parameters determine the local geometry (angle and shearing) at every edge, hence completely describe the polyhedron. A small perturbation in the shape parameter subject to (1) and (2) corresponds to a deformation of P into another weakly ideal polyhedron. Indeed, the convexity is stable under a small perturbation, then (1) and (2) guarantee that the hyperbolic vertex figures are closed polygonal curves, hence they are vertex figures of a weakly inscribed polyhedron.
6.4. Rigidity with respect to dihedral angles. We now have the necessary tools to prove the infinitesimal rigidity of weakly ideal polyhedra with respect to their dihedral angles.
Proposition 6.4. Let P ∈ P p,q andṖ be an infinitesimal deformation of P within P p,q . IfṖ does not change the dihedral angles Θ(P ) at first order, thenṖ is trivial.
Proof. LetṖ ∈ C p+q be an infinitesimal deformation of P , represented by the velocity of the vertices in CP 1 . Letτ = (τ e ) e∈E ∈ C |E| be the corresponding first-order variation of the shape parameters associated to the edges. Suppose thatṖ does not change the dihedral angles of P (at first order). This means that for all e ∈ E,τ e /τ e is real, because the argument of τ e is equal to the dihedral angle at the corresponding edge. Now consider the first-order variation iτ = (iτ e ) e∈E of the shape parameters. A crucial observation is that, since the conditions (1) and (2) above are polynomial, iτ again corresponds to a first-order deformation of P , which we can call iṖ . Now for all e ∈ E , iτ e /τ e is imaginary. This means that in the first-order deformation iṖ , the shear along the edges remains fixed (at first order). So iṖ does not change, at first order, the HS-structure induced on P .
By Proposition 6.3, iṖ is trivial, and it follows that the infinitesimal deformationṖ is also trivial. The result follows. 7. Topology 7.1. Ideal polyhedra. In this section we will conclude that Θ and ∆ are homeomorphisms. The first step is to prove that the domain P p,q is connected.
We work in a projective chart inconsistent with H 3 , in which ∂H 3 is the quadric of equation The following lemma allows us to place any weakly (p, q)-ideal polyhedron in a convenient position.
Lemma 7.1. For any P ∈ P p,q , there is an isometry T of H 3 such that T (P ) contains the origin and the points (0, 0, ±1).
Proof. The proof uses the Hyperplane Separation Theorem for hyperbolic space. We sketch here a quite standard proof in the spirit of [BV04], as some ingredient in this proof would be useful for us.
Note that H 3 ∩ P consists of two disjoint components, denoted by P + and P − , both are convex subsets of H 3 . Let u ∈ P + and v ∈ P − such that the hyperbolic distance between u and v achieves the minimum hyperbolic distance between P + and P − . This distance is necessarily finite, hence u and v are necessarily on the boundary ∂P . We claim that the hyperbolic plane H that perpendicularly bisects the segment uv separates P + and P − , i.e. P ± are on different sides of H. To see this, assume u ∈ P + is on the same side of H as P − . Then a perturbation of u towards u would be closer to v, contradicting our choice of u and v. Now let T be the isometry of H 3 that sends the separating plane H to infinity. Then the polar point of H, which is contained in P , is sent to the origin in the interior of T (P ). Moreover, the line through u and v is sent to the x 2 -axis. In particular, the points (0, 0, ±1) are also in the interior of T (P ).
Proposition 7.2. P p,q is connected.
Proof. Let P ∈ P p,q . Thanks to the previous lemma, we may assume that P contains the origin and the points (0, 0, ±1). We now define a deformation of P ∈ P p,q .
If v ∈ V ± , define v t , t > 1, as follows: v t = v if the x 2 -coordinate of v is smaller than t; otherwise, v t is a point of height x 2 = ±t obtained by moving v along the gradient of f (for metric induced on the quadric by the Euclidean metric dx 2 0 + dx 2 1 + dx 2 2 in a chart) towards (0, 0, ±1). We claim that the point set V t = {v t | v ∈ V + ∪ V − } remains in convex position for all t > 1. If v t ∈ V t is at height x 2 = ±t, v t is on the circle x 2 0 + x 2 1 = t 2 − 1; the convexity at v t is then immediate. Otherwise, v t coincides with a vertex v of P , and we claim that a supporting hyperplane H v of P at v is also a supporting hyperplane of P t . This can be seen by noting that, for any u ∈ ∂H 3 ± on the same side of H v as (0, 0, ±1), as long as u is sufficiently close to H v , the gradient of f at u points away from H v . Because (0, 0, ±1) ∈ P and H v is supporting, no vertex v t would move across H v as t decreases.
Define P t as the convex hull of V t . We see that P t = P for t sufficiently large. As t approaches 1, the vertices of P t lie, eventually, on two horizontal planes x 2 = ±t. Now assume another polyhedra P ∈ P p,q , which also contains the origin and the points (0, 0, ±1). For t sufficiently close to 1, both P t and P t have vertices on the horizontal planes x 2 = ±t. Polyhedra with vertices on these two planes form a connected subset of P p,q ; indeed, any choice of p and q ideal points on these two planes determines uniquely such a polyhedra. Hence we find a continuous path from between P and P , which proves the connectedness of P p,q .
Let P i p,q denote the open subset of P p,q consisting of polyhedra with an edge 1 + i − . {P i p,q | 1 ≤ i ≤ q} form an open cover of P p,q . During the deformation P t in the previous proof, we may rotate V − around the x 2 -axis so that 1 + i − remains an edge. This shows that Proposition 7.3. P i p,q is connected.
7.2. Admissible graphs. Correspondingly, let A i p,q denote the open subset of A p,q consisting of graphs θ with 1 + i − as an edge (recall the vertex labeling from Section 2). That is, either θ(1 + i − ) < 0, or θ(1 + i − ) = 0 but the graph remains polyhedral if we include 1 + i − as an edge. Then {A i p,q | 1 ≤ i ≤ q} form an open cover of A p,q .
Proposition 7.4. For 1 ≤ i ≤ q, A i p,q is homeomorphic to R 2(p+q−3) . We treat two cases separately.
Proof for the case p < 2 < q. We may take i = 1 without loss of generality. To construct a graph θ ∈ A 1 p,q , we first assign positive weights then negative weights. For the positive weights, our task is to find an outerplanar graph θ + on the vertices of the q-cycle with only positive edge weights. For the final result to be an element of A 1 p,q , we need the support of θ + to contain the q-cycle, and the sum of the weights of θ + to be π. This condition on the sum can be seen by adding up the weight sum of θ around all the vertices of the q-cycle, and noticing that the edges of θ + are double counted. Let A + be the set of all θ + satisfying those conditions.
Note that θ + can be embedded in the plane in such a way that the q-cycle is embedded as a q-gon, and the other edges are embedded as non-crossing diagonals of the q-gon. Let A + 0 denote the set of positive graphs that are only supported on the q-cycle, and A + 1 the set of positive graphs that are only supported on non-crossing diagonals. Then any θ + ∈ A + can be written in the form θ + = (1 − t)θ + 0 + tθ + 1 where θ + 0 ∈ A 0 , θ + 1 ∈ A 1 , for some 0 ≤ t < 1 (note the strict inequality!). It is easy to see that A + 0 is a (q − 1)-dimensional open simplex. Graphs in A + 1 with the same combinatorics (that is, supported on the same diagonals) also form an open simplex, whose dimension is the cardinality of their support minus 1. A + 1 is therefore a simplicial complex: The maximal simplices are of dimension q − 4, corresponding to the maximal set of non-crossing diagonals, and they are glued along their faces corresponding to common subsets. This simplicial complex is well-known as the boundary of a polyhedron, namely the associahedron [Lee89].
Therefore, the closure of A + = (1 − t)A + 0 + tA + 1 , 0 ≤ t < 1, is topologically the join of a (q − 1)-ball and a (q − 4)-sphere, hence homeomorphic to a (2q − 4)-ball. The openness of A + follows from the openness of A + 0 and the strict inequality t < 1. Once positive weights are assigned, the negative weights are uniquely determined since p = 1 and all vertices of the q-cycle are connected to only one negatively weighted edge. Hence A i p,q is homeomorphic to R 2(p+q−3) .
We need more ingredients for the case 2 ≤ p ≤ q. First, we claim that if a graph θ is admissible, and the negative weights of θ sum up to −2ω > −2π, then for any 0 < t < π/ω, the scaled graph tθ is also admissible. The claim follows from the following lemma, which guarantees that Condition (A1) is not violated after the scaling: Lemma 7.5. Let −2ω > −2π be the sum of negative weights of θ. Then any negative weight of θ is at least −ω.
Proof. We argue by contradiction and suppose that there is an edge e with negative weight strictly less than −ω. Let v be an endpoint of e. The sum of the positive weights on the red edges e 1 , · · · , e k adjacent to v is at least ω. Let v 1 , · · · , v k be their endpoints different from v; e is not adjacent to any of them. Then the sum of the negative weights over the blue edges adjacent to v 1 , · · · , v k must be strictly less than −ω. We then conclude that the sum of negative weights is not −2ω as assumed, but strictly less.
This lemma also proves the redundancy of Condition (A3) and part of Condition (A1). Any weight function that satisfy Conditions (W1) and (W2) can be normalized to satisfy Conditions (A1)-(A3). Hence these conditions are not present in the main Theorem 1.4.
Proof for the case 2 ≤ p ≤ q. We may take i = 1 without loss of generality. Fix a number 0 < ω < π. We will prove that the set of admissible graphs in A 1 p,q with negative weights summing up to −2ω is homeomorphic to R 2(p+q)−7 . To construct such a graph θ, we follow the same strategy as before. That is, first assign positive weights then negative weights.
For the positive weights, we need to find θ + that is the disjoint union of two outerplanar graphs, one on the vertices of the p-cycle, and the other on the vertices of the q-cycle, with only positive edge weights. Moreover, we need the sum of the weights of each disjoint component to be ω. Hence each component can be obtained by taking the θ + constructed for the case p < 2 < q, and multiplying its weights by a constant ω/π. The space of such θ + is then homeomorphic to R 2(p+q−4) .
We then propose an algorithm that assigns weights to negative edges and outputs an admissible graph in A 1 p,q . This algorithm depends on one parameter taken in a segment, hence proves the proposition.
Recall that vertices are labeled by 1 + , . . . , p + and 1 − , . . . , q − respectively, following the boundary of the outerplanar subgraphs in a compatible direction. Also recall that the weight w v of a vertex v is the sum of weights θ(u, v) over all other vertices u. The vertex weights change as we update the edge weights. Before we proceed, the weights are positive on every vertex, because only positive weights are assigned. Our goal is to cancel them with negative weights. We also keep track of two indices i and j, initially both 1. At each step, we assign a negative weight to the edge i + j − . Moreover, the graph will be embedded in S 2 throughout the algorithm.
We start with an embedding of θ + in S 2 , such that the two outerplanar components are embedded as two disjoint polygons with non-crossing diagonals. Interiors of the two polygons are declared as forbidden area: during our construction, no new edge is allowed to intersect this area. In other words, we are only allowed to draw edges within a belt.
For the first step, we draw a curve connecting 1 + and 1 − , to which we are free to assign any non-positive weight ranging from − min(w 1 + , w 1 − ) to 0. We also have the freedom to choose a sign σ = ±, and increment i if σ = +, increment j if σ = −.
In the following steps, we adopt a greedy strategy. We draw a curve connecting i + and j − , to which we assign the weight − min(w i + , w j − ). After this assignment, the face bounded by i + j − and the previously assigned edge is considered as a forbidden area for later construction: no new edge is allowed to intersect this area. Then we increment i if w i + = 0, and increment j if w j − = 0. If both weights vanish, we increment both indices.
Eventually, we will have i = p + 1 and j = q + 1, and the weights vanish at all vertices. Then the algorithm stops. The result is by construction the embedding of a (p, q)-admissible graph.
In this algorithm, being greedy is not only good, but also necessary. Note that the weights between vertices of smaller indices are already fixed. If we choose any bigger negative weight for the edge i + j − , then both w i + and w j − remain positive. They both need to be connected to a vertex with larger index to cancel the weight. This is however not possible, since neighborhoods of these vertices have been declared as forbidden area.
The algorithm is parametrized only by the two choices at the first step: a non-positive weight and a sign. The space of parameters is therefore homeomorphic to a segment.
We have Θ −1 (A i p,q ) ⊆ P i p,q . Let Θ i denote the restriction of Θ on P i p,q ; it is a covering map with images in A i p,q . Since P i p,q is connected and A i p,q simply connected, we conclude that Θ i p,q is a homeomorphism. This proves that the covering number of Θ is 1, i.e. Θ is a homeomorphism. 7.3. Admissible HS structures. Let H 2 denote the complete, simply connected hyperbolic surface with one cone singularity of angle . We extend this notation, and use H 2 0 for the hyperbolic surface with one cusp. A non-degenerate boundary component of dS 2 can be identified to the boundary of H 2 .
Let B be a subset of p points on ∂H 2 , considered up to isometries of H 2 . Let G B, be the space of CPP maps from ∂H 2 to ∂H 2 , 0 < < 2π, up to isometries of both H 2 and H 2 , with positive break point set B. We denote by G B the union of G B, for 0 < < 2π, i.e. set of all CPP maps on ∂H 2 with positive break point set B, up to isometries.
A horocyclic p-gon is the intersection of p horodisks in H 2 . Figure 8 shows a horocyclic 3-gon and a horocyclic 4-gon.
The key observation is the following homeomorphism η from G B to the space of horocyclic p-gons bounded by horocycles based at B.
Label the elements of B as b 1 , . . . , b p = b 0 in the clockwise order in the disk model. They are the vertices of an ideal p-gon P ⊂ H 2 . Consider a map γ ∈ G B, . It maps P to an ideal p-gon P ⊂ H 2 with a cone singularity of angle in its interior. The vertices of P are γ(b i ). Shrinking h 4 truncates the 3-gon into the 4-gon on the right. We also sketch a scaling of the 4-gon, which is the key in the proof of Proposition 7.10.
We then obtain a horocyclic p-gon η(γ) as follows. Cut P into p triangles by the geodesics from γ(b i ) to s, the cone singularity of H 2 . Then fold each triangle γ(b i )sγ(b i+1 ) inward, isometrically, into the triangle b i s i b i+1 in H 2 . Let h i be the horocycle based at b i passing through s i . Then η(γ) is the intersection of the horodisks bounded by the h i 's.
Since γ preserves horocycles, s i−1 must also lie on h i . Hence the horocyclic segments s i s i+1 form a piecewise horocyclic closed curve, denoted by h. For x ∈ ∂H 2 and y ∈ H 2 , we use ξ(x, y) to denote the other ideal end of the geodesic that emerges from x and passes through y. Define a map g : ∂H 2 → ∂H 2 such that g −1 (b i ) = {ξ(b i , y) | y ∈ h i ∩h} and, for some x between b i and b i+1 , g −1 (x) = ξ(x, s i ). The map g is continuous and monotone, and has the property that x / ∈ g −1 (x), hence its degree must be 1. This proves that h is embedded, hence the boundary of η(γ).
Conversely, given a horocyclic polygon bounded by horocycles h i based at b i ∈ B, let s i ∈ h i ∩h i+1 be its vertices. Then we can glue the triangles b i s i b i+1 into an ideal p-gon with a cone singularity of angle . More specifically, is the sum of the angles ∠b i s i b i+1 . This proves that η is a homeomorphism.
Remark 7.7. It is interesting to note from the horocyclic polygons that < 2π. Let s be a point in the interior of the horocyclic polygon. We then have ∠b i s i b i+1 < ∠b i sb i+1 for all i. Yet the sum of ∠b i sb i+1 is equal to 2π.
Proposition 7.8. G B is homeomorphic to R |B| .
Proof. The proof is by induction on the cardinality p = |B|. Up to isometries of H 2 , we may consider B fixed.
For p = 2, a horocyclic 2-gon P 2 is bounded by two horocycles. It is determined by the position of an intersection point of the two horocycles. This point can be chosen arbitrarily in H 2 , proving the statement for p = 2. Now consider a horocyclic (p − 1)-gon P p−1 bounded by horocycles h 1 , . . . , h p−1 based at b 1 , . . . , b p−1 ∈ B, and let s i ∈ h i ∩ h i+1 and s p−1 ∈ h p−1 ∩ h 1 denote the vertices of P p−1 . We now construct a horocyclic p-gon P bounded by horocycles with bases in B. For this, it suffices to choose a horocycle h p that truncates the vertex s p−1 of P p−1 . This can be done by taking the horocycle h p at b p passing through s p−1 , then shrinking it. On the left of Figure 8 we illustrate a truncation of a 3-gon into a 4-gon. We can continue to shrink h p until it hits another vertex of P p−1 .
Hence G B is homeomorphic to G B\{bp} × R, which is R |B| by induction.
In the following we will consider the closures of G B . The boundary ∂G B consists of three parts, namely G B,0 , G B,2π , and ∂G B, , 0 < < 2π. We now define and describe these spaces.
We use the notation G B,0 for the space of CPP maps from ∂H 2 to ∂H 2 0 , up to isometries of both H 2 and H 2 0 , with positive break point set B. As before, we can interpret a map γ ∈ G p,0 as gluing the boundary of an ideal p-gon P ⊂ H 2 to the boundary of an ideal p-gon P ⊂ H 2 0 , where P contains a cusp s. We triangulate P by connecting its vertices to s, and triangulate P arbitrarily, and glue them through γ to obtain a triangulation T of the 2-sphere. The shearing coordinates on the p − 3 interior edges of P are determined by the positions of the break points. The shearing coordinates on the remaining 2p edges of T are governed by the condition that the shearing around each vertex of T should sum up to 0. Since T has p + 1 vertices, we conclude that G B,0 is homeomorphic to R p−1 = R |B|−1 .
The part ∂G B, , 0 < < 2π, consists of CPP maps from ∂H 2 to ∂H 2 up to isometries of both H 2 and H 2 , with break point set B ⊂ B and marked projective point set B \ B . The homeomorphism η extends continuously to this boundary. More specifically, let p = |B| and p = |B |. Then for γ ∈ ∂G B, , η(γ) is a horocyclic p -gons P bounded by horocycles with bases in B , together with p − p additional horocycles with bases in B \ B "supporting" P (that is, they intersect ∂P but disjoint from the interior of P ). The left side of Figure 8 is an example with p = 4 and p = 3. We also abuse the notation G p,2π for the space of projective homeomorphisms from ∂H 2 to itself up to isometries of H 2 , with a set of marked points B. In fact, the marking here is superficial; hence G p,2π consists of a single element.
Let δ i denote the distance from s i to the geodesic b i b i+1 . The hyperbolic triangle formula shows that α i and δ i are related by the formula cosh δ i sin α i /2 = 1. We now deform the horocyclic p-gon by moving every s i , simultaneously, along the geodesic perpendicular to b i b i+1 , to a new position s i . The following lemma is the crucial observation for the proof that follows later.
Lemma 7.9. If the deformation is performed in such a way that the ratio cosh δ i / cosh δ i is the same for every i, then there is a horocycle h i passing through every adjacent pair s i−1 and s i .
Proof. Let k be the common ratio of cosh δ i / cosh δ i . We use the half-plane model of H 2 , and assume that b i = ∞. The situation is illustrated in Figure 9. Let ψ i be the (Euclidean) angle ∠s i b i+1 b i−1 and ψ i−1 be the angle ∠s i−1 b i−1 b i+1 . Then for j = i or i − 1, we can calculate the distances (see for instance [And05, §3.5]) δ j = ln |csc ψ j + cot ψ j | = arccosh csc ψ j . This is particularly convenient because we have cosh δ j sin ψ j = 1. Hence cosh δ j / cosh δ j = k implies that sin ψ j / sin ψ j = 1/k for every i. In the half-plane model, s i and s i−1 are moving along the circles centered at b i+1 and b i−1 , respectively, such that their heights are both scaled by 1/k. Therefore, their new positions s i−1 and s i are again at the same height, hence on the same horocycle h i based at b i = ∞.
This deformation is sketched on the right side of Figure 8. We see from Figure 9 that, by moving s j 's outwards the horocyclic polygon, one can multiply cosh(δ i ) by an arbitrarily large constant. However, if we move s j 's inwards the horocyclic polygon, s i and s i−1 would eventually merge.
Proposition 7.10. G B, , 0 ≤ ≤ 2π, are contractible, and homeomorphic to R |B|−1 if < 2π. Proof. Through any given point γ ∈ G B , the deformation described above defines a continuous path with monotonically changing cone angle . We use as the parameter and denote this path by c γ ( ). See Figure 10 for an illustration.
To decrease , one needs to move the vertices outwards the horocyclic polygon. We have seen that, in this direction, one can travel along c γ ( ) until hitting G B,0 .
To increase , one needs to move the vertices inwards the horocyclic polygon. In this direction, however, the path c γ ( ) would in general hit some γ ∈ ∂G B, for some < 2π, as shown in Figure 10. Further movement of the vertices in the same direction would destroy the p-gon. However, we can continue deforming γ as a gluing map in G B for some B ⊂ B. Hence c γ ( ) is extended into the closure of G B , along which one can increase up to 2π; see Figure 10.
This path is uniquely defined through every γ ∈ G B , and two path do not intersect inside G B ; intersection is only possible on the boundary. For 0 ≤ = ≤ 2π, let f , : G B, → G B, be the continuous map that sends γ ∈ G B, to γ = c γ ( ) ∈ G B, . Then f , and f , define a homotopy equivalence between G B, and G B, . Consequently, G B, are all of the same homotopy type. We have seen that G B,0 and G B,2π are contractible, and so must be G B, for 0 < < 2π.
In general, f , and f , are not inverse to each other. However, if the path emerging from γ ∈ G B, arrives at γ = f , (γ) ∈ G B, without hitting the boundary of G B , then one can travel backwards along the same path. The reversed path defines f , , hence we have f , • f , (γ) = f , (γ ) = γ. This is the case, in particular, when > and γ ∈ G B, . Then f , defines a homeomorphism between G B, and its image f , (G B, ) ⊂ G B, . We finally conclude that G B, , 0 < < 2π, by its continuity in , are all homeomorphic to G B,0 , thus to R |B|−1 .
It is now clear that G B is foliated by G B, , 0 < < 2π, as illustrated in Figure 10. An admissible HS structure is obtained by gluing copies of H 2 to the ideal boundary components of dS 2 . We now conclude on the topology of M p,q , and prove Theorem 3.4.
Case 2 ≤ p ≤ q. Condition (C2) requires a cycle with alternating colors; we call such a cycle an alternating cycle. As in the previous case, the existence of an alternating cycle that visits every edge implies that every vertex is adjacent to a blue edge, but the converse is not true. However, we have the following equivalence, which does not depend on Condition (C1).
Proposition 8.2. If the edges of a graph are colored in blue and red, then Condition (C2) is equivalent to (C"2) Every edge belongs to an alternating cycle (which does not necessarily visits every edge).
The proof is immediate from the composition and decomposition of cycles.
Proof of Theorem 1.3 when 2 ≤ p ≤ q. If w has an alternating cycle c, the number of visits defines a graph supported on the edges of c, which we denote by w c . We can choose a positive number α such that w = w − αw c is supported on a proper subgraph of w, but still satisfy Condition (W1) on other edges. Most importantly, w satisfy Condition (W2) because both w and w c do. The cycle c no longer exist in w . We repeat this operation if there are other alternating cycles. Since the graph is finite, we will obtain a graphw with no alternating cycle in finitely many steps.
If an edge e 0 of w does not belong to any alternating cycle, it must also be an edge of w. But we prove in the following that this is not possible.
Assume that e 0 is red and let v + 0 and v − 0 be its vertices. Condition (W2) guarantees that v + 0 is adjacent to a blue edge, e 1 , whose other vertex is denoted by v + 1 . The same argument continues and we obtain an alternating path e 0 , e 1 , e 2 , . . . .
Since the graph is finite, this path will eventually intersect itself. That is, v + i = v + i for some 0 ≤ i < i (note that we don't consider v − 0 as visited by e 0 ). If e i and e i are of the same color, e i +1 , . . . , e i form an alternating cycle, contradicting our assumption. Hence e i and e i must have different colors.
The same argument applies on the other vertex v − 0 of e 0 . We obtain an alternating path e 0 , e −1 , e −2 , . . . . Let v − j denote the common vertex of e −j and e −j−1 . This path eventually intersect itself, i.e. v − j = v − j for some 0 ≤ j < j (this time we don't consider v + 0 as visited by e 0 ). Again, e −j and e −j must have different colors.
Remark 8.3. The feasibility region specified in Theorem 1.4 is a polyhedral cone. The proof above shows that the extreme rays of this cone correspond to the minimal alternating cycles in the graph. Example 8.4. The example in Figure 12 shows that Condition (C2) is essential. This graph is not the 1-skeleton of any weakly ideal polyhedron with the inner square in H 3 + and the outer square in H 3 − . A fairly elementary argument, left to the reader, shows that there is no alternating cycle containing edge e. This can also be shown using Theorem 1.4, since if a graph w satisfies Conditions (W1) and (W2), we would have w(a) + w(b) < w(a ) + w(b ) , w(c) + w(d) < w(c ) + w(d ) , from which a contradiction immediately follows.
Remark 8.5. Given a graph G with edges colored in blue and red, we define a directed graphG as follows: • Each vertex v of G lifts to two vertices v + and v − inG.
• Each red edge uv of G lifts to two oriented edges u − v + and v − u + inG.
• Each blue edge uv of G lifts to two oriented edges u + v − and v + u − inG. It is quite clear from the definition that an alternating cycle in G lifts to two oriented cycle inG, and any oriented cycle inG projects to an alternating cycle in G. Hence finding an alternating cycle in G is equivalent to finding an oriented cycle inG. The latter can be solved by a simple depth-or breath-first search.