Related papers: Hollow polytopes of large width
We give a new combinatorial proof for the number of convex polyominoes whose minimum enclosing rectangle has given dimensions. We also count the subclass of these polyominoes that contain the lower left corner of the enclosing rectangle…
For natural numbers $n$ and $l > d \geq 2$, let $ES_d(l,n)$ be the minimum $N$ such that any set of at least $N$ points in $\mathbb{R}^d$ contains either $l$ points contained in a common $(d-1)$-dimensional hyperplane or $n$ points in…
A convex body $R$ in $\mathbb R^d$ is called reduced if the minimal width $\Delta(R')$ of each convex body $R'\subset R$ different from $R$ is strictly smaller than the minimal width $\Delta(R)$ of $R$. In this article we construct a…
Lattice-free sets (convex subsets of $\mathbb{R}^d$ without interior integer points) and their applications for cutting-plane methods in mixed-integer optimization have been studied in recent literature. Notably, the family of all integral…
It is proven here that the diameter of the d-dimensional associahedron is 2d-4 when d is greater than 9. Two maximally distant vertices of this polytope are explicitly described as triangulations of a convex polygon, and their distance is…
It is known that polytopes with at most two nonsimple vertices are reconstructible from their graphs, and that $d$-polytopes with at most $d-2$ nonsimple vertices are reconstructible from their 2-skeletons. Here we close the gap between 2…
Let $d$ be a nonnegative integer, and let $P \subset \mathbb R^d$ be a $d$-dimensional convex lattice polytope. In this article, we prove that the ratio of the volume of a normal-sized miniature of $P$ to that of $P$ is $1:\binom{2d+1}{d},$…
We consider how flat a lattice simplex contained in the hypercube $[0,k]^d$ can be. This question is related to the notion of kissing polytopes: two lattice polytopes contained in the hypercube $[0,k]^d$ are kissing when they are disjoint…
We study the maximum weight convex polytope problem, in which the goal is to find a convex polytope maximizing the total weight of enclosed points. Prior to this work, the only known result for this problem was an $O(n^3)$ algorithm for the…
In this short note we give a construction of an infinite series of Delone simplices whose relative volume grows super-exponentially with their dimension. This dramatically improves the previous best lower bound, which was linear.
Denote by ${\mathcal K}^d$ the family of convex bodies in $E^d$ and by $w(C)$ the minimal width of $C \in {\mathcal K}^d$. We ask for the greatest number $\Lambda_n ({\mathcal K}^d)$ such that every $C \in {\mathcal K}^d$ contains a…
By using representation theory, we reduce the size of the set of possible values for the dimension of the convex hull of all feasible points polytope of an orthogonal array (OA) defining integer linear program (ILP). Our results address the…
Let $\mathcal{P} \subset \mathbb{R}^d$ be a lattice polytope of dimension $d$. Let $b$ denote the number of lattice points belonging to the boundary of $\mathcal{P}$ and $c$ that to the interior of $\mathcal{P}$. It follows from a lower…
A polytope $D$ whose vertices belong to a lattice of rank $d$ is Delaunay if there is a circumscribing $d$-dimensional ellipsoid, $E$, with interior free of lattice points so that the vertices of $D$ lie on $E$. If in addition, the…
The mean width is a measure on three-dimensional convex bodies that enjoys equal status with volume and surface area [Rota]. As the phrase suggests, it is the mean of a probability density f. We verify formulas for mean widths of the…
A lattice $(d,k)$-polytope is the convex hull of a set of points in $\mathbb{R}^d$ whose coordinates are integers ranging between $0$ and $k$. We consider the smallest possible distance $\varepsilon(d,k)$ between two disjoint lattice…
A small polygon is a polygon of unit diameter. The maximal perimeter and the maximal width of a convex small polygon with $n=2^s$ sides are unknown when $s \ge 4$. In this paper, we propose an approach to construct convex small $n$-gons of…
The Hirsch Conjecture (1957) stated that the graph of a $d$-dimensional polytope with $n$ facets cannot have (combinatorial) diameter greater than $n-d$. That is, that any two vertices of the polytope can be connected by a path of at most…
This paper proves the following results: Besides parallelograms and centrally symmetric hexagons, there is no other convex domain which can form a two-, three- or four-fold lattice tiling in the Euclidean plane. If a centrally symmetric…
A theorem of Scott gives an upper bound for the normalized volume of lattice polygons with exactly $i>0$ interior lattice points. We will show that the same bound is true for the normalized volume of lattice polytopes of degree 2 even in…