Related papers: On the integer points in a lattice polytope: n-fol…

Let $K \subseteq \mathbb{R}^d$ be a convex body and let $\mathbf{w} \in \operatorname{int}(K)$ be an interior point of $K$. The coefficient of asymmetry $\operatorname{ca}(K,\mathbf{w}) := \min\{ \lambda \geq 1 : \mathbf{w} - K \subseteq…

In 2021, Hibi et. al. studied lattice points in $\mathbb{N}^2$ that appear as $(\depth R/I,\dim R/I)$ when $I$ is the edge ideal of a graph on $n$ vertices, and showed these points lie between two convex polytopes. When restricting to the…

We investigate the average number of lattice points within a ball for the $n$th cyclotomic number field, where the lattice is chosen at random from the set of unit determinant ideal lattices of the field. We show that this average is nearly…

Let $d$ be a positive integer. For a finite set $X \subseteq \mathbb{R}^d$, we define its integer cone as the set $\mathsf{IntCone}(X) := \{ \sum_{x \in X} \lambda_x \cdot x \mid \lambda_x \in \mathbb{Z}_{\geq 0} \} \subseteq \mathbb{R}^d$.…

We derive tight expressions for the maximum number of $k$-faces, $0\le{}k\le{}d-1$, of the Minkowski sum, $P_1\oplus{}P_2$, of two $d$-dimensional convex polytopes $P_1$ and $P_2$, as a function of the number of vertices of the polytopes.…

Fakhruddin has proved that for two lattice polygons P and Q any lattice point in their Minkowski sum can be written as a sum of a lattice point in P and one in Q, provided P is smooth and the normal fan of P is a subdivision of the normal…

Every polyhedron can be decomposed into a Minkowski sum (or vector sum) of a bounded polyhedron and a polyhedral cone. This paper establishes similar statements for some classes of discrete sets in discrete convex analysis, such as…

The main purpose of this note is to prove an upper bound on the number of lattice points of a centrally symmetric convex body in terms of the successive minima of the body. This bound improves on former bounds and narrows the gap towards a…

We classify the unimodular equivalence classes of inclusion-minimal polygons with a certain fixed lattice width. As a corollary, we find a sharp upper bound on the number of lattice points of these minimal polygons.

Let S be a finite subset of a field. For multivariate polynomials the generalized Schwartz-Zippel bound [2], [4] estimates the number of zeros over Sx...xS counted with multiplicity. It does this in terms of the total degree, the number of…

This paper is a survey of recent advances as well as open problems in the study of face numbers of centrally symmetric simplicial polytopes and spheres. The topics discussed range from neighborliness of centrally symmetric polytopes and the…

We continue our study of ends of non-compact manifolds, with a focus on the inward tameness condition. For manifolds with compact boundary, inward tameness, has significant implications. For example, such manifolds have stable homology at…

This article derives closed-form parametric formulas for the Minkowski sums of convex bodies in d-dimensional Euclidean space with boundaries that are smooth and have all positive sectional curvatures at every point. Under these conditions,…

In this article we introduce the study of the number of pairs of non-comparable elements in a distributive lattice $\L$. We give several tight lower and upper bounds for the number and give as an application the lattices precisely for which…

Suppose $G$ is a locally solid lattice group. It is known that there are non-equivalent classes of bounded homomorphisms on $G$ which have topological structures. In this paper, our attempt is to assign lattice structures on them. More…

We present a generic and systematic approach for constructing D-dimensional lattice models with exactly solvable d-dimensional boundary states localized to corners, edges, hinges and surfaces. These solvable models represent a class of…

The lambda-dilate of a set A is lambda*A={lambda a : a \in A}. We give an asymptotically sharp lower bound on the size of sumsets of the form lambda_1*A+...+lambda_k*A for arbitrary integers lambda_1,...,lambda_k and integer sets A. We also…

We show that the largest possible diameter $\delta(d,k)$ of a $d$-dimensional polytope whose vertices have integer coordinates ranging between $0$ and $k$ is at most $kd-\lceil2d/3\rceil$ when $k\geq3$. In addition, we show that…

A well known result by Lagarias and Ziegler states that there are finitely many equivalence classes of d-dimensional lattice polytopes having volume at most K, for fixed constants d and K. We describe an algorithm for the complete…

Given a filter $\Delta$ in the poset of compositions of $n$, we form the filter $\Pi^{*}_{\Delta}$ in the partition lattice. We determine all the reduced homology groups of the order complex of $\Pi^{*}_{\Delta}$ as ${\mathfrak…