On counterexamples in questions of unique determination of convex bodies

We discuss a construction that gives counterexamples to various questions of unique determination of convex bodies.


Introduction
This note is motivated by the following question from the book "Geometric tomography" by R. J. Gardner [G,Prob. 7.6,p. 289].
Question 1.1. Let K and L be origin-symmetric convex bodies in R 3 such that the sections K ∩ H and L ∩ H have equal perimeters for every 2-dimensional subspace H of R 3 . Is it true that K = L?
The problem has attracted a lot of attention, but at the moment it is still open; see [HNRZ], [R], [Y] for some partial results. The question can also be generalized to higher dimensions and all intermediate intrinsic volumes V i , as well as sections of other dimensions.
In this note we show that the question has a negative answer in the class of general (not necessarily symmetric) convex bodies.
One of the natural steps in generalizing questions involving sections is to replace sections by slabs (or "thick sections"). We discuss such problems at the end of the paper.
Throughout the paper we use the following notation. We write S n−1 and B n 2 for the unit sphere and closed unit ball in Euclidean space R n , and G(n, k) for the Grassmanian of all k-dimensional subspaces of R n . The hyperplane through the origin with normal vector ξ ∈ S n−1 is denoted by ξ ⊥ . A convex body in R n is a compact convex set with nonempty interior. In this paper we always assume that convex bodies contain the origin in their interiors. The orthogonal projection of a convex body K onto a subspace V is denoted by K|V . If Q is a convex body in an i-dimensional subspace V ⊆ R n , 1 ≤ i ≤ n, then we write vol i (Q) for its volume. For a set C in R n , its reflection in the origin is denoted by −C. We say that a set C is origin symmetric if C = −C. Two sets C and D in R n are said to be congruent if there is an isometry T of R n such that T (C) = D.
For a convex body K in R n its support function h K is defined by Key words and phrases. Convex bodies, sections, projections, slabs, intrinsic volumes. First author supported in part by U.S. National Science Foundation Grants DMS-0652684 and DMS-1101636. Second author supported in part by NSERC.
Clearly, h K is positively homogeneous of degree 1, and therefore is determined by its values on the unit sphere. It is also easy to see that for any subspace V of R n the support function of K|V , as a convex body in V , is just the restriction of h K to V .
If K is a compact set containing the origin in its interior and star-shaped with respect to the origin, then its radial function ρ K is defined by x ∈ S n−1 .
We say that a convex body K is of class C ∞ + if its boundary is a C ∞ manifold in R n with positive Gaussian curvature at each point.
Let K be a compact convex set in R n . Its intrinsic volumes V i (K), 1 ≤ i ≤ n, can be defined via Steiner's formula where the addition is the Minkowski addition, κ n−i is the volume of the (n − i)dimensional Euclidean ball, and ǫ ≥ 0. In particular, if K is a convex body in R n , then V n (K) is its volume, and V n−1 (K) is half the surface area. For these and other facts about intrinsic volumes we refer the reader to the book [S].

Results
The idea of the construction that we use in this note goes back to Gardner and Volčič [GV] and Goodey, Schneider and Weil [GSW].
First we prove an auxiliary lemma.
Lemma 2.1. Let K and L be convex bodies in R n containing the origin in their interiors. The following properties are equivalent.
In the second case, let ρ K (ξ) > ρ L (ξ), say. Consider the ray {x ∈ R n : x = tξ, t ≥ 0}. It intersects the boundaries of L and K at some points, which we will denote by p and q correspondingly. Since K \ L = −(L \ K), the ray {x ∈ R n : x = −tξ, t ≥ 0} intersects L \ K in the interval (−p, −q]. Hence, the points −p and −q are on the boundaries of L and K correspondingly. This implies that ρ K (ξ) = ρ L (−ξ) and For the reader's convenience, we provide a short proof of the following proposition. Examples of this type appeared in [G,Thm 3.3.17,Thm 3.3.18] (see also [GV]).
Proposition 2.2. There exist noncongruent convex bodies K and L satisfying the properties of Lemma 2.1. Moreover, the bodies can chosen to be C ∞ + bodies of revolution, or polytopes.
Proof. 1) K and L are C ∞ + bodies of revolution. We start with two nonzero infinitely smooth functions φ and ψ, supported in the intervals (1/3 − δ, 1/3 + δ) and (2/3 − δ, 2/3 + δ) correspondingly, where 0 < δ < 1/6. Next, for t ∈ [−1, 1] and ǫ > 0, we define . Finally, we choose ǫ sufficiently small to guarantee that f and g are concave on [−1, 1]. Now we consider two convex bodies in R n , The boundaries of K and L are generated by rotating the graphs of the functions f and g defined above. Since these generating curves are not congruent, it follows that K and L are not congruent.
By construction, K and L are of class C ∞ , and also K \ L = −(L \ K). Finally, choosing ǫ > 0 sufficiently small, ρ K − ρ B n 2 C 2 (S n−1 ) and ρ L − ρ B n 2 C 2 (S n−1 ) can be made as small as we wish. By formula (0.41) from [G] this is enough to ensure that K and L have strictly positive curvature.
2) K and L are polytopes. Let a 1 ,...,a n be distinct positive numbers and consider the rectangular box R = [−a 1 , a 1 ] × · · · × [−a n , a n ]. Take any two vertices u and v of R that are connected by an edge. The idea is that K will be constructed by cutting off vertices u and v from R, and L by cutting off vertices u and −v.
More precisely, let F and H be supporting hyperplanes to R at the vertices u and v correspondingly, such that R ∩ F = {u} and R ∩ H = {v}. Suppose F is given by x, ξ = t for some ξ ∈ S n−1 and t > 0, and H is given by x, η = s for some η ∈ S n−1 and s > 0. If λ > 0 is sufficiently small, then the sets R∩{x : x, ξ ≥ t−λ} and R ∩ {x : x, η ≥ s − λ} do not overlap and contain no vertices of R other than u and v correspondingly. Define The bodies K and L are not congruent, since the vertices u and v that we cut off from R belong to the same edge, while this is not true for u and −v.
Finally, observe that K and L satisfy the property (i) of Lemma 2.1. Lemma 2.1 and Proposition 2.2 lead to the following result; see Theorems 3.3.17, 3.3.18, 6.2.18, 6.2.19 in [G].
Corollary 2.3. There are noncongruent convex bodies K and L in R n such that ∀ξ ∈ S n−1 , vol n−1 (K|ξ ⊥ ) = vol n−1 (L|ξ ⊥ ) This result remains true if the (n − 1)-dimensional volume of the projections is replaced by their i-th intrinsic volumes. See Gardner [G,Thm 3.3.17,Thm 3.3.18] for i = 1, n − 1, and Goodey, Schneider and Weil [GSW] for all other cases. Now we return to Question 1.1 stated in the introduction. The next theorem shows that without the assumption of origin-symmetry, this question would have a negative answer.
Theorem 2.4. There are noncongruent convex bodies K and L in R n such that for all i and k with 1 ≤ i ≤ k ≤ n − 1 we have for all H ∈ G(n, k).
Moreover, K and L can be constructed in such a way that both of them are C ∞ + bodies of revolution, or both K and L are polytopes.
Then, for every H ∈ G(n, k), the convex bodies K ∩ H and L ∩ H in the subspace H satisfy the relation By Lemma 2.1, for these k-dimensional bodies we also have and, in particular, the last relation is true for the projections of K ∩ H and L ∩ H onto i-dimensional subspaces of H, 1 ≤ i ≤ k. We apply Lemma 2.1 to get vol i ((K ∩ H)|V ) = vol i ((L ∩ H)|V ) for every i-dimensional subspace V of H. By Kubota's integral recursion (see [S, p.295]), we have for some constant c i,k . Here, by G(H, i) we mean the Grassmanian of all i-dimensional subspaces of H. Thus, The following result is proved in [Y]. Let k be an integer with 2 ≤ k ≤ n − 1. If K and L are origin-symmetric convex polytopes in R n such that V k−1 (K ∩ H) = V k−1 (K ∩ H) for all H ∈ G(n, k), then K = L. Theorem 2.4 shows that this result is false without the symmetry assumption.
It is interesting to note that the following result is true.
Proposition 2.5. Let i = j be integers, 1 ≤ i, j ≤ n − 1, and let K and L be convex bodies (containing the origin in their interiors) such that Then for all k, 1 ≤ k ≤ n − 1, we have (3) vol k (K|H) = vol k (L|H), ∀H ∈ G(n, k).
Clearly, if K and L are two convex bodies such that all their corresponding idimensional projections have the same i-volume, and all corresponding j-dimensional projections have the same j-volume, then one can not expect that the sections of K and L will have equal volumes (simply because sections are not invariant under translations). Under the same conditions, is it possible to translate K and L so that all their sections would have the same volumes? The answer is again negative. Goodey, Schneider and Weil [GSW,p. 86] remark that there are convex polytopes K and L such that (3) holds for all k, but none of their translates satisfy K \ L = −(L \ K). Thus, as the proof of Proposition 2.5 shows, no translates of K and L can satisfy (1) and (2) for some i = j.
Our next question concerns slabs in convex bodies. For t > 0 and ξ ∈ S n−1 consider the slab of width 2t perpendicular to ξ defined by There are various problems related to volumes of slabs in convex bodies; see e.g. [BK], [KK]. Here we suggest the following.
Question 2.6. Let K and L be origin-symmetric convex bodies in R n that contain the Euclidean ball of radius t in their interiors. Suppose that for some i (1 ≤ i ≤ n) ∀ξ ∈ S n−1 .
Is it true that K = L?
Our next result shows that the previous question has a negative answer if we drop the assumption of origin-symmetry.
Theorem 2.7. There are noncongruent convex bodies K and L in R n that contain the Euclidean ball of radius t > 0 in their interiors and such that V i (K ∩ S t (ξ)) = V i (L ∩ S t (ξ)), ∀ξ ∈ S n−1 , for all 1 ≤ i ≤ n. Moreover, both K and L can be chosen to be C ∞ + bodies of revolution or polytopes, as in Theorem 2.4.
Proof. Consider the same pair of bodies K and L as used in Theorem 2.4. Without loss of generality, we may assume that t > 0 is small enough so that tB n 2 is contained in the interior of both K and L. Since S t (ξ) = −S t (ξ), it follows that where for brevity we write S = S t (ξ). Now, as in Theorem 2.4, we see that all idimensional projections of K ∩ S and L ∩ S have the same i-volumes, and Kubota's integral recursion finishes the proof.
Since intrinsic volumes are continuous (in the Hausdorff metric) valuations on convex sets, it follows that Theorem 2.4 can be derived from Theorem 2.7. It is enough to notice that K ∩ S t (ξ) approaches K ∩ ξ ⊥ in the Hausdorff metric.