Elementary moves on lattice polytopes

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Abstract

We introduce a graph structure on the set of Euclidean polytopes. The vertices of this graph are the d-dimensional polytopes contained in Rd and its edges connect any two polytopes that can be obtained from one another by either inserting or deleting a vertex, while keeping their vertex sets otherwise unaffected. We prove several results on the connectivity of this graph, and on a number of its subgraphs. We are especially interested in several families of subgraphs induced by lattice polytopes, such as the subgraphs induced by the lattice polytopes with n or n+1 vertices, that turn out to exhibit intriguing properties.

Introduction

In this paper, we introduce a graph structure on the set of Euclidean polytopes. The vertices of this graph are the d-dimensional polytopes contained in Rd and its edges connect two polytopes when they can be obtained from one another by a transforming move, that we want to keep as elementary as possible. Here, by elementary, we mean that these moves should preserve as much as possible the combinatorics of a polytope's boundary. We will consider two types of moves, an insertion move and a deletion move. If x is a point in RdP, an insertion move will transform P into the convex hull of P{x}. Note that x is then necessarily a vertex of the resulting polytope. If v is a vertex of P, a deletion move will transform P into the convex hull of V{v}, where V is the vertex set of P. Without any other requirement, these moves cannot be considered elementary in the above sense, as they can alter the combinatorics of the boundary complex of P significantly. Indeed, the convex hull of P{x} can have fewer (and possibly many less) vertices that P itself. This happens, for instance, when the convex hull of P{x} contains at least two vertices of P in its relative interior. An undesirable consequence is that deletion moves would then not be the inverse of insertion moves as P would not always be recovered by deleting x from the convex hull of P{x}. A natural way to solve this issue consists in allowing an insertion move only when all the vertices of P remain vertices of the polytope resulting from that insertion.

Definition 1.1

Consider a d-dimensional polytope P contained in Rd and denote by V its set of vertices. A point xRd can be inserted in P if the convex hull of P{x} admits V{x} as its vertex set. A vertex vV can be deleted from P when the convex hull of V{v} is d-dimensional.

By this definition, deletion moves and insertion moves are now the inverse of one another. Recall that all the polytopes we consider here are full-dimensional, making the requirement that deletion moves do not decrease the dimension of a polytope necessary. In particular, a vertex v of a polytope P can be deleted from P if and only if P is not a pyramid with apex v over a (d1)-dimensional polytope. Consider the graph whose vertices are the d-dimensional polytopes contained in Rd and whose edges connect two polytopes that can be obtained from one another by an insertion move (or a deletion move). This graph, which we will refer to as Γ(d) here, has an uncountable number of vertices and, as soon as d2, its vertices all have uncountable degree. Indeed, consider an arbitrary point x in the boundary of a polytope P, distinct from any vertex of P. One can insert in P any point outside of P that is close enough to x.

The purpose of this paper is to investigate the connectivity of a number of subgraphs of Γ(d). As all the vertices of Γ(1) are isolated, it will be assumed throughout the paper that d is at least 2. Our first main result, proven in Section 2, deals with the subgraphs induced in Γ(d) by the polytopes with n or n+1 vertices.

Theorem 1.2

The polytopes with n or n+1 vertices induce a connected subgraph of Γ(d) whose diameter is at least 4nd and at most 6n4.

The connectedness of Γ(d) itself will be obtained as a consequence of Theorem 1.2. Note that Γ(d) provides a metric on the set of d-dimensional polytopes, in a very different spirit than, for instance the Gromov-Hausdorff distance [19]: instead of measuring how far two bodies are from being isometric, we measure how long it takes to build them from one another with operations that affect as little as possible the combinatorics of their boundary. More generally, we consider certain well-defined classes of polytopes as geometric spaces, in a similar manner to what has recently been done with flip operations for the triangulations of a topological surface [26] or with projective and normal transformations for the realization space of the hypercube [2].

The two families of graphs we mostly focus on are the subgraph Λ(d) induced in Γ(d) by the lattice polytopes and the subgraph Λ(d,k) induced in Λ(d) by the polytopes contained in the hypercube [0,k]d, where k is a positive integer. Here, by a lattice polytope we mean a polytope whose vertices all belong to the lattice Zd. These polytopes pop up in many places in the mathematical literature as, for instance in combinatorial optimization [24], [25], [28], in discrete geometry [5], [10], [23], or in connection with toric varieties [9], [17]. It is noteworthy that, in the particular case of lattice polytopes, alternative deletion and insertion moves have been considered, that amount to change the number of lattice points contained in a polytope by exactly one [7], [8], [11]. These alternative moves can affect the combinatorics of a polytope's boundary in an arbitrary way, but they provide a natural way to enumerate the lattice polytopes that contain a fixed number of lattice points [7], [8]. In contrast, our moves can be used to enumerate the lattice polytopes with a fixed number of vertices contained in a given hypercube.

Observe that Λ(d) is a highly non-regular graph: it admits both vertices with finite degree and vertices with infinite (but countable) degree. In particular, Λ(d) gathers in a coherent metric structure, polytopes whose boundaries exhibit dramatically different behaviors regarding the ambient lattice. For instance, there are lattice polytopes, such as the cube [0,1]d, in which no lattice point can be inserted, while for the lattice simplices, no deletion move is possible. The graph Λ(d,k) is particularly relevant to the study of the lattice polytopes contained in [0,k]d, that have attracted significant attention [1], [4], [13], [14], [15], [22]. Our second main result deals with a subgraph of Λ(d,k).

Theorem 1.3

The subgraph induced in Λ(d,k) by the simplices and the polytopes with d+2 vertices is connected.

This theorem will be proven in Sections 3 and 4 along with a number of its consequences. For instance, it is proven in Section 3 that some lattice point in [0,k]d can always be inserted in a d-dimensional lattice simplex contained in [0,k]d. As we shall see, the proof of this seemingly straightforward statement alone turns out to be surprisingly involved. The connectedness of Λ(d) and Λ(d,k) will both be obtained from Theorem 1.3 in Section 4. Observe that the latter connectedness result allows for the definition of a Markov chain whose states are the d-dimensional lattice polytopes contained in [0,k]d, and whose stationary distribution is uniform [12].

In Section 5, we will study the number of lattice points that can be inserted in, or removed from a lattice polytope. We will describe a family of d-dimensional lattice polytopes contained in the hypercube [0,k]d, where d and k can grow arbitrarily large, such that every lattice point in [0,k]d can be either inserted or deleted. These polytopes belong to the broader family of the empty lattice polytopes, that is widely studied and is interesting in its own right [3], [6], [16], [20], [21], [27], [28]. We will also exhibit lattice polytopes with arbitrarily large dimension and number of vertices such that no insertion of a lattice point is possible. As an immediate consequence, the subgraph induced in Λ(d) by the polytopes with n or n+1 vertices is not always connected.

In particular we obtain the following in Section 5.

Theorem 1.4

The subgraph induced in Λ(2) by the polygons with n or n+1 vertices is disconnected when n is distinct from 3 and 5.

It is a consequence of Theorem 1.3 that triangles and quadrilaterals induce a connected subgraph of Λ(2), which settles the first exception in the statement of Theorem 1.4. We settle the other exception in Section 6 as follows.

Theorem 1.5

Pentagons and hexagons induce a connected subgraph of Λ(2).

In order to prove Theorem 1.5, we will study a particular connected component of the subgraph induced in Λ(2) by the polygons with n or n+1 vertices. The proof of Theorem 1.5 consists in showing that this connected component is the whole subgraph when n is equal to 5. We conclude the article in Section 7 by asking a number of questions. A part of these questions arises from the intriguing behavior of the subgraphs induced in Λ(d) by the polytopes with n or n+1 vertices. We will also mention a number of other subgraphs of Γ(d), whose study may be interesting.

Section snippets

The connectivity of Γ(d)

In this section we investigate the connectedness of Γ(d) itself and of its subgraphs induced by the polytopes with n and n+1 vertices, where n is greater than d. We also obtain precise bounds on the diameter of these subgraphs.

Consider a d-dimensional polytope P contained in Rd. We will denote by aff(F) the affine hull of a face F of P. If F is a facet, then aff(F) is a hyperplane of Rd and in this case, we denote by HF(P) the closed half-space of Rd bounded by aff(F) such that PHF(P)=F. For

The insertion move for lattice simplices

Connecting two polytopes within Λ(d) turns out to be much more complicated than within Γ(d). Recall that Lemma 2.1 makes it obvious that an insertion move is always possible on a polytope when the vertices of this polytope are not constrained to belong to a lattice. Indeed, as already mentioned, one can always insert a point in the polytope, provided this point is close enough to the boundary of the polytope but far enough from its vertices. In the case of lattice polytopes, inserting a point

The connectedness of Λ(d) and Λ(d,k)

We begin the section by proving that Λ(2,k) is connected. This will serve as the base case for the inductive proof that Λ(d,k) is connected. We call corner simplex of [0,k]d the simplex whose vertices are the origin of Rd (the point whose all coordinates are zero), and the d lattice points in [0,k]d at distance 1 from the origin.

Lemma 4.1

For any positive integer k, the subgraph induced in Λ(2,k) by the triangles and the quadrilaterals is connected.

Proof

Since each vertex of a quadrilateral can be deleted, we

The number of possible insertion and deletion moves

The main purpose of this section is to study how the vertex degrees in Λ(d) and Λ(d,k) decompose between insertion and deletion moves. In particular, we will exhibit a family of polytopes whose dimension and number of vertices can be arbitrarily large, but in which no lattice point can be inserted. As a consequence, the vertex degrees in Λ(d) can be finite. We then turn our attention to Λ(d,k). The vertex degrees in this graph are bounded above by (k+1)d, the number of lattice points in [0,k]d.

The subgraph of Λ(2) induced by pentagons and hexagons

According to Theorem 1.4, the subgraph induced in Λ(2) by the polygons with n or n+1 vertices is always disconnected except possibly when n is equal to 3 and 5. By Theorem 4.3, this subgraph is connected when n is equal to 3. In this section, we deal with the remaining case. As a first step, we describe a large connected component of the subgraph induced in Λ(2) by the polygons with n or n+1 vertices.

We will make use of the following notion.

Definition 6.1

A polygon P will be called oblique if it admits two

Discussion and open problems

We have introduced a graph structure on the set of the d-dimensional polytopes contained in Rd. We have proven, among other things, that this graph is connected, as well as its subgraph induced by the lattice polytopes. The distances in this graph provide a measure of dissimilarity on polytopes in terms of how long it takes to transform two of them into one another by a sequence of elementary moves. This also allows to gather in a coherent metric structure very different objects from both the

Acknowledgements

Lionel Pournin is partially supported by the ANR project SoS (Structures on Surfaces), grant number ANR-17-CE40-0033.

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