Related papers: Classification of Convex lattice polytopes
We introduce the multi-width of a lattice polytope and use this to classify and count all lattice tetrahedra with multi-width $(1,w_2,w_3)$. The approach used in this classification can be extended into a computer algorithm to classify…
The aim of this paper is to study alcoved polytopes, which are polytopes arising from affine Coxeter arrangements. This class of convex polytopes includes many classical polytopes, for example, the hypersimplices. We compare two…
A classification of SL$(n)$ invariant valuations on the space of convex polytopes in $R^n$ without any continuity assumptions is established. A corresponding result is obtained on the space of convex polytopes in $R^n$ that contain the…
After giving a short introduction on smooth lattice polytopes, I will present a proof for the finiteness of smooth lattice polytopes with few lattice points. The argument is then turned into an algorithm for the classification of smooth…
We give an explicit classification of translation-invariant, Lorentz-invariant continuous valuations on convex sets. We also classify the Lorentz-invariant even generalized valuations.
Preorder polytopes, defined from preorders on finite sets, are introduced and studied from a lattice point enumeration point of view. They naturally generalize arbor polytopes, recently introduced and studied by the second named author.…
We introduce the notion of a polyptych lattice, which encodes a collection of lattices related by piecewise linear bijections. We initiate a study of the new theory of convex geometry and polytopes associated to polyptych lattices. In…
A complete classification of the $\delta$-vectors of lattice polytopes whose normalized volumes are at most $4$ is known. In the present paper, we will classify all the $\delta$-vectors of lattice polytopes with normalized volumes $5$.
We survey results devoted to the lattice of varieties of monoids. Along with known results, some unpublished results are given with proofs. A number of open questions and problems are also formulated.
We classify the three-dimensional lattice polytopes with two interior lattice points. Up to unimodular equivalence there are 22,673,449 such polytopes. This classification allows us to verify, for this case only, a conjectural upper bound…
We provide a complete classification up to isomorphism of all smooth convex lattice 3-polytopes with at most 16 lattice points. There exist in total 103 different polytopes meeting these criteria. Of these, 99 are strict Cayley polytopes…
There are (at least) two reasons to study random polytopes. The first is to understand the combinatorics and geometry of random polytopes especially as compared to other classes of polytopes, and the second is to analyze average-case…
In this paper, we present the classification of generalized Wallach spaces and discuss some related problems.
The lattice size of a lattice polytope is a geometric invariant which was formally introduced in the context of simplification of the defining equation of an algebraic curve, but appeared implicitly earlier in geometric combinatorics.…
For a d-dimensional convex lattice polytope P, a formula for the boundary volume is derived in terms of the number of boundary lattice points on the first $\floor{d/2}$ dilations of P. As an application we give a necessary and sufficient…
We propose some problems on the classification of toric manifolds from the viewpoint of topology and survey related results.
We describe the computation of polytope volumes by descent in the face lattice, its implementation in Normaliz, and the connection to reverse-lexicographic triangulations. The efficiency of the algorithm is demonstrated by several high…
A permutation polytope is the convex hull of a group of permutation matrices. In this paper we investigate the combinatorics of permutation polytopes and their faces. As applications we completely classify permutation polytopes in…
Zonotopes are a rich and fascinating family of polytopes, with connections to many areas of mathematics. In this article we provide a brief survey of classical and recent results related to lattice zonotopes. Our emphasis is on connections…
The symmetry approach is used for classification of integrable isotropic vector Volterra lattices on the sphere. The list of integrable lattices consists mainly of new equations. Their symplectic structure and associated PDE of vector…