For a polyhedron \(P\) let \(B(P)\) denote the polytopal complex that is formed by all bounded faces of \(P\). If \(P\) is the intersection of \(n\) halfspaces in \(\mathbb R ^D\), but the maximum dimension \(d\) of any face in \(B(P)\) is much smaller, we show that the combinatorial complexity of \(P\) cannot be too high; in particular, that it is independent of \(D\). We show...

We prove a conjecture of Ambrus, Ball and Erdélyi that equally spaced points maximize the minimum of discrete potentials on the unit circle whenever the potential is of the form $$\begin{aligned} \sum _{k=1}^n f(d(z,z_k)), \end{aligned}$$where \(f:[0,\pi ]\rightarrow [0,\infty ]\) is non-increasing and convex and \(d(z,w)\) denotes the geodesic distance between z and w on the...

We prove small-deviation estimates for the volume of random convex sets. The focus is on convex hulls and Minkowski sums of line segments generated by independent random points. The random models considered include (Lebesgue) absolutely continuous probability measures with bounded densities and the class of log-concave measures.

A geometric \(k\)-configuration is a collection of points and straight lines in the plane so that \(k\) points lie on each line and \(k\) lines pass through this point. We introduce a new construction method for constructing \(k\)-configurations with non-trivial dihedral or chiral (i.e., purely rotational) symmetry, for any \(k \ge 3\); the configurations produced have \(2^{k-2...

It is proved that if we partition a d-dimensional cube into \(n^d\) small cubes and color the small cubes in \(m+1\) colors then there exists a monochromatic connected component consisting of at least \(f(d, m) n^{d-m}\) small cubes. Another proof of this result is given in Matdinov’s preprint (Size of components of a cube coloring, arXiv:1111.3911, 2011)

We prove that if a pure simplicial complex \(\Delta \) of dimension \(d\) with \(n\) facets has the least possible number of \((d-1)\)-dimensional faces among all complexes with \(n\) faces of dimension \(d\), then it is vertex decomposable. This answers a question of J. Herzog and T. Hibi. In fact, we prove a generalization of their theorem using combinatorial methods.

Let \(n\) be a positive integer, not a power of two. A Reinhardt polygon is a convex \(n\)-gon that is optimal in three different geometric optimization problems: it has maximal perimeter relative to its diameter, maximal width relative to its diameter, and maximal width relative to its perimeter. For almost all \(n\), there are many Reinhardt polygons with \(n\) sides, and many...

Given a family F of n pairwise disjoint compact convex sets in the plane with non-empty interiors, let T(k) denote the property that every subfamily of F of size k has a line transversal, and T the property that the entire family has a line transversal. We illustrate the applicability of allowable interval sequences to problems involving line transversals in the plane by proving...

We consider the Ehrhart h ∗-vector for the hypersimplex. It is well-known that the sum of the \(h_{i}^{*}\) is the normalized volume which equals the Eulerian numbers. The main result is a proof of a conjecture by R. Stanley which gives an interpretation of the \(h^{*}_{i}\) coefficients in terms of descents and exceedances. Our proof is geometric using a careful book-keeping of...

In this paper we discuss some affine properties of convex equal-area polygons, which are convex polygons such that all triangles formed by three consecutive vertices have the same area. Besides being able to approximate closed convex smooth curves almost uniformly with respect to affine length, convex equal-area polygons admit natural definitions of the usual affine differential...

Let m 1,…,m d be positive integers, and let G be a subgroup of ℤ d such that m 1ℤ×⋯×m d ℤ⊆G. It is easily seen that if a unit cube tiling [0,1) d +t,t∈T, of ℝ d is invariant under the action of G, then for every t∈T, the number |T∩(t+ℤ d )∩[0,m 1)×⋯×[0,m d )| is divisible by |G|. We give sufficient conditions under which this number is divisible by a multiple of |G|. Moreover, a...

The skeleton of a polyhedral set is the union of its edges and vertices. Let \(\mathcal {P}\) be a set of fat, convex polytopes in three dimensions with n vertices in total, and let f max be the maximum complexity of any face of a polytope in \(\mathcal {P}\). We prove that the total length of the skeleton of the union of the polytopes in \(\mathcal {P}\) is at most O(α(n)⋅log∗ n...

Quite recently, an Orlicz Minkowski problem has been posed and the existence part of this problem for even measures has been presented. In this paper, the existence part of the Orlicz Minkowski problem for polytopes is demonstrated. Furthermore, we obtain a solution of the Orlicz Minkowski problem for general (not necessarily even) measures.

We explore the art gallery problem for the special case that the domain (gallery) P is an m-polyomino, a polyform whose cells are m unit squares. We study the combinatorics of guarding polyominoes in terms of the parameter m, in contrast with the traditional parameter n, the number of vertices of P. In particular, we show that \(\lfloor\frac{m+1}{3} \rfloor\) point guards are...

A graph G is a k-sphere graph if there are k-dimensional real vectors v 1,…,v n such that ij∈E(G) if and only if the distance between v i and v j is at most 1. A graph G is a k-dot product graph if there are k-dimensional real vectors v 1,…,v n such that ij∈E(G) if and only if the dot product of v i and v j is at least 1. By relating these two geometric graph constructions to...

We present approximation algorithms for maximum independent set of pseudo-disks in the plane, both in the weighted and unweighted cases. For the unweighted case, we prove that a local-search algorithm yields a PTAS. For the weighted case, we suggest a novel rounding scheme based on an LP relaxation of the problem, which leads to a constant-factor approximation.Most previous...

We show that the Delaunay triangulation of a set of n points distributed nearly uniformly on a p-dimensional polyhedron (not necessarily convex) in d-dimensional Euclidean space is \(O(n^{\frac{d-k+1}{p}})\), where \(k = \lceil\frac{d+1}{p+1} \rceil\). This bound is tight in the worst case and improves on the prior upper bound for most values of p.

Given a function f on a surface and a tolerance δ>0, we construct a function f δ subject to ‖f δ −f‖∞≤δ such that f δ has a minimum number of critical points. Our construction relies on a connection between discrete Morse theory and persistent homology and completely removes homological noise with persistence ≤2δ from the input function f. The number of critical points of the...

Let P be a set of points in general position in the plane. Join all pairs of points in P with straight line segments. The number of segment-crossings in such a drawing, denoted by \(\operatorname {cr}(P)\), is the rectilinear crossing number of P. A halving line of P is a line passing through two points of P that divides the rest of the points of P in (almost) half. The number of...

We study the structure of the space of diametrically complete sets in a finite dimensional normed space. In contrast to the Euclidean case, this space is in general not convex. We show that its starshapedness is equivalent to the completeness of the parallel bodies of complete sets, a property studied in Moreno and Schneider (Isr. J. Math. 2012, doi: 10.1007/s11856-012-0003-6...

We consider the Traveling Salesman Problem with Neighborhoods (TSPN) in doubling metrics. The goal is to find a shortest tour that visits each of a collection of n subsets (regions or neighborhoods) in the underlying metric space. We give a quasi-polynomial time approximation scheme (QPTAS) when the regions are what we call α-fat weakly disjoint. This notion combines the existing...

We consider the problem of fitting a subspace of a specified dimension k to a set P of n points in ℝ d . The fit of a subspace F is measured by the L τ norm, that is, it is defined as the τ-root of the sum of the τth powers of the Euclidean distances of the points in P from F, for some τ≥1. Our main result is a randomized algorithm that takes as input P, k, and a parameter 0<ε<1...