# The finiteness threshold width of lattice polytopes

## Abstract

In each dimension there is a constant such that for every all but finitely many lattice with -polytopes lattice points have lattice width at most We call . the *finiteness threshold width* in dimension and show that .

Blanco and Santos determined the value Here, we establish . This implies, in particular, that there are only finitely many empty . of width larger than two. (This last statement was claimed by Barile et al. in [Proc. Am. Math. Soc. 139 (2011), pp. 4247–4253], but we have found a gap in their proof.) -simplices

Our main tool is the study of lifts of hollow -dimensional -polytopes.

## 1. Introduction

*Lattice polytopes* are convex polytopes with vertices in (or in any other lattice). They appear in combinatorics, algebraic geometry, symplectic geometry, optimization, or statistics and have applications to mathematical physics in string theory. In particular, considerable effort has gone into several classification projects for several classes of them, with motivation stemming from different sources. For example:

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A monumental task and now a shining example is the classification of reflexive polytopes up to dimension by Kreuzer and Skarke Reference KS00, the data for these and other Calabi-Yau manifolds can be found online under http://hep.itp.tuwien.ac.at/~kreuzer/CY.html.

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Smooth reflexive polytopes were classified up to dimension by Øbro Reference Øb07 and in dimension by Lorenz and Paffenholz Reference LP08 (see also https://polymake.org/polytopes/paffenholz/www/fano.html). This classification led to new discoveries about smooth reflexive polytopes in arbitrary dimension and hereby helped solving long-open problems Reference AJP14Reference LN15Reference NP11Reference OSY12.

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Lattice polytopes with a single lattice point in their interior (assumed to be the origin) are important in algebraic geometry. They correspond to projective toric varieties with at most

*canonical singularities*, which is why they are called*canonical polytopes*. Canonical polytopes all of whose boundary lattice points are vertices are called*terminal*. Canonical lattice polytopes were fully enumerated by Kasprzyk -dimensionalReference Kas10. The data for this and a lot more can be found in the graded ring database (http://www.grdb.co.uk).- •
A classification especially useful for us is that of

*hollow*polytopes, by which we mean lattice polytopes without interior lattice points. In dimension two their list consists of the polygons of width one plus the second dilation of a unimodular triangle. In dimension three the full classification has recently been completed by Averkov et al. Reference AWW11 and Reference AKW17. See Section 5 for details.- •
We call

*empty*a (necessarily hollow) lattice polytope with no lattice point apart from its vertices. Empty simplices are of special interest, since they are the building blocks into which every lattice polytope can be decomposed, and since they correspond to terminal quotient singularities in algebraic geometry. Their classification in dimension three is by now classical Reference Whi64. Their classification in dimension four has been completed recently Reference IS21, after efforts coming both from algebraic geometry Reference BBBK11Reference Bob09Reference MMM88Reference San90 and discrete geometry Reference HZ00. See Remark 1.5 for more details.

All these classifications are modulo *unimodular equivalence*, sometimes called We say that two lattice polytopes are -isomorphism.*unimodularly equivalent*, or just *equivalent*, if there is a lattice-preserving affine isomorphism mapping them onto each other.

From the point of view of discrete geometry alone, it seems natural to classify, or enumerate, *all* lattice polytopes of a given dimension and with a certain number of lattice points. We call the latter the *size* of a lattice polytope. In dimension this is trivial, since the unique lattice segment of size is that of length In dimension . it is also easy, at least from the theoretical point of view: Pick’s Theorem implies that lattice polygons of size have area smaller than which in turn implies that they can be unimodularly transformed to fit in , Hence, there are finitely many of them and they can in principle be enumerated by brute force. However, in dimension . and higher the task is a-priori undoable, since the number is infinite already for the smallest possible case, that of *empty tetrahedra* (that is, lattice of size -polytopes Indeed, the following infinite family of so-called ).*Reeve tetrahedra* was described more than 60 years ago Reference Ree57:

Still, Blanco and Santos Reference BS16a found a way of making sense of the question in dimension They proved that, for each . all but finitely many lattice , of size -polytopes have width one. They also enumerated lattice polytopes of width larger than one and of sizes up to eleven Reference BS16aReference BS16bReference BS18.

Here, the width of a lattice polytope *with respect to a linear functional* is defined as

and the *lattice width*, or simply *width*, of the polytope is the minimum such where ranges over non-zero integer functionals:

For example, has width one if and only if it lies between two consecutive lattice hyperplanes.

The starting point in this paper is the observation that the finiteness result of Blanco and Santos generalizes as follows:

For each we call *finiteness threshold width* in dimension the minimum constant such that for every the number of lattice of size -polytopes and width larger than is finite.

For instance, since, as said above, there are only finitely many lattice of each size. Blanco and Santos’ aforementioned result states that -polytopes Our main result is the exact value of . :

That is, for each . there are only finitely many lattice of size -polytopes and width greater than .

This implies the following result:

There are infinitely many empty of width two but only finitely many of larger width. -simplices

Haase and Ziegler Reference HZ00, Proposition 6 found infinitely many empty of width -simplices . implies there are only finitely many of larger width.

■The second part of Corollary 1.4 is the main result in Barile et al. Reference BBBK11, but we have found out that the proof given in that paper is incomplete. Indeed, the authors use a classification of infinite families of empty of width -simplices that had been conjectured to be complete by Mori et al. Reference MMM88 and proved by Sankaran Reference San90 and Bover Reference Bob09, * for simplices whose determinant (i.e., their normalized volume) is a prime number*. But when the determinant is not prime other infinite families do arise, such as the following explicit example: the empty with vertices -simplices , , , and where the determinant , is a multiple of and coprime with As a conclusion, the proof of Corollary .1.4 given in Reference BBBK11 is valid only for simplices of prime determinant.

We thank O. Iglesias for the computations leading to finding this (and other) families and we thank the authors of Reference BBBK11 for acknowledging (private communication) their mistake and for helpful discussions about the extent of it.

After the present paper was completed, a new proof of Corollary 1.4 has been obtained by Iglesias and Santos, which gives the following more explicit information: there are exactly empty of width larger than two, all of width three except for one of width four -simplicesReference IS19. Furthermore, Reference IS21 contains the full classification of empty including the additional infinite families of width two that arise for nonprime determinant. -simplices,

Our bounds on come from relating it to the maximum widths of hollow and/or empty As already mentioned, a lattice polytope is -polytopes.*hollow* if there is no lattice point in its interior and *empty* if its vertices are the only lattice points it contains.

We denote and the maximum widths of hollow and empty respectively. -polytopes,

Finiteness of (and hence of is usually called the “flatness theorem”, dating back to Khinchine (1948); see, e.g., )Reference KL88. The current best upper bound of for some constant (used in the proof of Theorem 1.1) is by Rudelson Reference Rud00, building on work by Banaszczyk et. al Reference BLPS99. As for lower bounds, follows from hollowness of the dilation of a unimodular -th while -simplex, was proved by Sebő Reference Seb99 by slightly modifying this same dilated to make it empty. -simplex

Along the paper, we prove the following properties and bounds of and where , is the stratification of the threshold width in terms of size. That is, is the minimal width such that there exist only finitely many lattice of size -polytopes and width Clearly, . and, in particular, each is finite.

None of the inequalities or (for is sharp, as the following table of known values shows. )

lower bounds | upper bound | |||

1 | ||||

2 | ||||

3 | ||||

4 | ||||

5 |

The values of , have been discussed above. For the rest: ,

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In dimension the unique hollow lattice segment is equivalent to , and then , .

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In dimension the second dilation of a unimodular triangle is the only hollow lattice polygon of width larger than one (see, e.g., ,Reference Tre08). Hence and, since this polygon is not empty, .

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In dimension Howe ( ,Reference Sca85, Thm. 1.3) proved that For . Averkov et al. ( ,Reference AWW11, Theorem 2.2 and Reference AKW17, Theorem 1) have classified all hollow and their maximum width is three (see more details in Lemma -polytopes5.3), so .

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In dimension Haase and Ziegler ,Reference HZ00 showed which implies , by part (4) of Theorem 1.7.

The structure of the paper is as follows. The monotonicity properties stated in parts (1) and (2) of Theorem 1.7 are proved at the beginning of Section 2. We then prove the upper bound (Lemma 2.3) from the following statement, which combines results of Hensley Reference Hen83, Lagarias–Ziegler Reference LZ91 and Nill–Ziegler Reference NZ11: all but finitely many lattice of bounded size are hollow and project to hollow -polytopes This fact implies that in order to find an infinite family of lattice -polytopes. of bounded size we can focus on lifts (see Definition -polytopes2.4) of hollow polytopes of one dimension less. The fact that all but finitely many lifts of a lattice are -polytope and have the same width (Theorem -dimensional2.7) then implies that in order to decided whether a lattice polytope has infinitely many lifts of bounded size it is enough to look at *tight* lifts, which are inclusion-minimal lifts of a polytope (see Definition 2.9 and Corollary 2.11).

In Section 3 we prove sufficient properties for hollow to have infinitely many lifts of bounded size. In particular, we prove the existence of such hollow -polytopes of widths -polytopes and which provides the lower bounds , (Corollary 3.5) and (Corollary 3.7). Moreover, we get the following characterization of the finiteness threshold width:

For all , equals the maximum width of a hollow lattice -polytope for which there are infinitely many (equivalence classes of) lattice -polytopes of bounded size projecting to .

One direction of the theorem is easy, since a as in the statement has all but finitely many of its lifts of the same width as (Theorem 2.7). The other is less obvious since might a priori be achieved by the existence of infinitely many hollow -polytopes each with finitely many lifts of size , .

In dimension the infinite family of Reeve tetrahedra are lifts of size , of a unit square, which is a hollow polygon of width one. On the other hand, the unique hollow polygon of width larger than one is the second dilation of the unimodular triangle, which has only finitely many lifts of bounded size (see Reference BS16a, Corollary 22). Hence .

In dimension observe that , follows from the fact that the following hollow of width two can be lifted to infinitely many empty -polytope (Haase and Ziegler -simplicesReference HZ00, Proposition 6):

Sections 4 and 5 are aimed at proving our main result (Theorem 1.3). By Theorem 1.9 and Example 1.10, it suffices to show that each hollow of width larger than two has finitely many -polytope lifts of bounded size. For this we first prove sufficient conditions for lattice polytopes (in arbitrary dimension) to have only finitely many lifts of bounded size (Section -dimensional4). Subsequently in Section 5 we apply this to the list of hollow of width larger than two. This list, containing only five polytopes, is derived from the classification of -polytopes*maximal* hollow by Averkov et al. ( -polytopesReference AWW11, Theorem 2.2 and Reference AKW17, Theorem 1).

In light of these results, we ask the following questions.

Besides the monotonicity in parts and of Theorem 1.7, does always hold? The case follows from Reference HZ00, Proposition 1: every empty is a facet of infinitely many empty -simplex of at least the same width. -simplices

For all known values ( we have ) That is, the finiteness threshold width for all lattice . is determined by empty -polytopes Does this hold for arbitrary -simplices. ?

## 2. Finiteness threshold width and lifts of hollow polytopes

### Monotonicity of the finiteness threshold widths

Parts (1) and (2) of Theorem 1.7 have the following proofs:

for all .

Let be such that there exists an infinite family of lattice of size -polytopes and width We are going to show that for each . there is a of size and width containing To prove this, let . be an integer functional giving width to and assume without loss of generality that , Taking any point . we easily get a of width and properly containing (see Figure 1). If has more than one lattice point, remove them one by one until only one remains (which can always be done; simply choose a vertex of not in and replace with the convex hull of then iterate). ;

That implies the lemma except for the fact that different polytopes and may produce isomorphic and so it is not obvious that , is an infinite family. But each element of can only correspond to *at most* elements from (because is recovered from by removing one of its lattice points), so the proof is complete.

for all , .

Let be such that, for some there is an infinite family , of lattice of size -polytopes and width Then, . is a family of of size -polytopes and width A priori two different . can give isomorphic polytopes in ’s but each polytope in , can correspond to only finitely many since ’s is the projection of along the direction of an edge. Hence is infinite and .

■The following lemma proves part (3) of Theorem 1.7:

Let All but finitely many lattice . of size bounded by -polytopes are hollow and admit a projection to some hollow lattice In particular, -polytope. for all .

As argued in the proof of Theorem 1.1, the number of non-hollow lattice of size bounded by -polytopes is finite. Hence, all but finitely many lattice of size bounded by -polytopes are hollow.

On the other hand, Nill and Ziegler Reference NZ11, Corollary 1.7 proved that all but finitely many hollow admit a projection to a hollow -polytopes And these have width at most that of their projection, which is -polytope. .

■### Finiteness threshold width via polytopes with infinitely many lifts of bounded size

We say that a lattice polytope is a *lift* of a lattice -polytope if there is a lattice projection with Without loss of generality, we will typically assume . to be the map that forgets the last coordinate.

Two lifts and with projections , and are *equivalent* if there is a unimodular equivalence with That is, if for each . , (the equivalence maps a point in the fiber of under to a point in the fiber of , under See Figure ).2 for examples of equivalent and non-equivalent lifts.

We say that “ has finitely many lifts of bounded size” if for every there are finitely many equivalence classes of lifts of of size Accordingly, .“ has infinitely many lifts of bounded size” means that there is an for which there are infinitely many equivalence classes of lifts of .

Saying that “ has infinitely many lifts of bounded size” is equivalent to saying that “there are infinitely many (equivalence classes of) lattice polytopes of bounded size that have a lattice projection to The implication from right to left is trivial, and the implication from left to right follows from the fact that once ”. is fixed there is a finite number of integer affine projections (an overestimate is where , and are the numbers of lattice points in and respectively; , is also an upper bound, since an affine map is determined by the image of an affine basis).

Our interest on these concepts comes from the following fact: for all , is *at least* the maximum width of a lattice -polytope that admits infinitely many lifts of bounded size (see Corollary 2.8). For its proof we need a couple of technical results about the dimension and the width of the lifts of a polytope.

A lift of may have the same dimension as and still not be unimodularly equivalent to it. For example, the segment in can be lifted to the primitive segment However, the number of different such lifts of . is finite, modulo the equivalence relation in Definition 2.4:

A polytope -dimensional has only finitely many lifts. -dimensional

Every lift -dimensional of can be described as follows: there is an affine map with

and such that is integer in all vertices of Assuming, without loss of generality, that . is linear and the origin is a vertex of this implies , where , is the lattice spanned by the vertices of Two such functionals give equivalent lifts if, and only if, they are in the same class modulo . Thus, the number of different lifts equals the index of . in .

■Let be a lattice of width -polytope Then all lifts . of have width All but finitely many of them have width . .

The first part of the statement is clear, since projecting cannot decrease width. It remains to show that only finitely many lifts of have width strictly smaller than which follows from the next claim: if a lift , of has width smaller than then

Finiteness of the volume of implies finitely many possibilities for .

To prove the volume bound, let be a lift of of width and let , be an integer functional attaining We have that . since implies Assume without loss of generality that . Then we have that . is contained in

which is a slanted prism projecting to and with every fiber of length (For the latter, observe that each fiber is a segment with endpoints . , for some and with Hence, the Euclidean volume of this slanted prism is ).

and, since is contained in it, we get that

where the middle inequality follows from being a nonzero integer.

We thank an anonymous referee for this proof, significantly simpler than the one we originally had.

■This gives us:

For all , is at least the maximum width of a lattice