Related papers: Lattice 3-polytopes with six lattice points

Abstract polytopes are a combinatorial generalization of convex and skeletal polytopes. Counting how many flag orbits a polytope has under its automorphism group is a way of measuring how symmetric it is. Polytopes with one flag orbit are…

We review 28 uniform partitions of 3-space in order to find out which of them have graphs (skeletons) embeddable isometrically (or with scale 2) into some cubic lattice ${\bf Z}_n$. We also consider some relatives of those 28 partitions,…

We show by a direct construction that there are at least $\exp\{cV^{(d-1)/(d+1)}\}$ convex lattice polytopes in $\mathbb{R}^d$ of volume $V$ that are different in the sense that none of them can be carried to an other one by a lattice…

We say a lattice tetrahedron whose centroid is its only non-vertex lattice point is lattice barycentric. The notation T(a,b,c) describes the lattice tetrahedron with vertices {0, e_1, e_2, a e_1 + b e_2 + c e_3}. Our result is that all such…

Minkowski's second theorem on successive minima asserts that the volume of a 0-symmetric convex body K over the covolume of a lattice \Lambda can be bounded above by a quantity involving all the successive minima of K with respect to…

The extension complexity of a polytope measures its amenability to succinct representations via lifts. There are several versions of extension complexity, including linear, real semidefinite, and complex semidefinite. We focus on the last…

We use the finite lattice method to count the number of punctured staircase and self-avoiding polygons with up to three holes on the square lattice. New or radically extended series have been derived for both the perimeter and area…

This paper concerns the number of lattice points in the plane which are visible along certain curves to all elements in some set S of lattice points simultaneously. By proposing the concept of level of visibility, we are able to analyze…

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 $5$.

Given a lattice polygon $P$ with $g$ interior lattice points, we associate to it the moduli space of tropical curves of genus $g$ with Newton polygon $P$. We completely classify the possible dimensions such a moduli space can have. For…

The contact polytope of a lattice is the convex hull of its shortest vectors. In this paper we classify the facets of the contact polytope of the Leech lattice up to symmetry. There are 1,197,362,269,604,214,277,200 many facets in 232…

Bosse et al. conjectured that for every natural number $d \ge 2$ and every $d$-dimensional polytope $P$ in $\real^d$ there exist $d$ polynomials $p_0(x),...,p_{d-1}(x)$ satisfying $P=\{x \in \mathbb{R}^d : p_0(x) \ge 0, >..., p_{d-1}(x) \ge…

We study the set of square-free parts of volume polynomials associated with four planar lattice polytopes. This is motivated by the problem of describing possible pairwise intersection numbers of four curves in $(\mathbb{C}^*)^2$ with…

We consider the lattice point problem corresponding to a family of elliptic paraboloids in $\mathbb{R}^d$ with $d\ge3$ and we prove the expected to be optimal exponent, improving previous results. This is especially noticeable for $d=3$…

In [Problems on polytopes, their groups, and realizations, Periodica Math. Hungarica 53 (2006) 231-255] Schulte and Weiss proposed the following problem: {\em Characterize regular polytopes of orders $2^np$ for $n$ a positive integer and…

We give a complete list of the points in the spectrum $$\mathcal{Z}=\{\inf_{(x,y)\in\Lambda,xy\neq0}{\left\vert xy\right\vert},\,\text{$\Lambda$ is a unimodular rational lattice of $\mathbb{R}^2$}\}$$ above $\frac{1}{3}.$ We further show…

We describe the geometry of an arrangement of 24-cells inscribed in the 600-cell. In $\S$7 we apply our results to the even unimodular lattice $E_8$ and show how the 600-cell transforms $E_8$/2$E_8$, an 8-space over the field $\bf{F}$$_2$,…

Orthogonal surfaces are nice mathematical objects which have interesting connections to various fields, e.g., integer programming, monomial ideals and order dimension. While orthogonal surfaces in one or two dimensions are rather trivial…

We study a combinatorial problem that recently arose in the context of shape optimization: among all triangles with vertices $(0,0)$, $(x,0)$, and $(0,y)$ and fixed area, which one encloses the most lattice points from $\mathbb{Z}_{>0}^2$?…

An empty simplex is a lattice simplex in which vertices are the only lattice points. We show two constructions leading to the first known empty simplices of width larger than their dimension: - We introduce cyclotomic simplices and…