Related papers: A New Slant on Lebesgue's Universal Covering Probl…

This paper views the honeycomb conjecture and the Kepler problem essentially as extreme value problems and solves them by partitioning 2-space and 3-space into building blocks and determining those blocks that have the universal extreme…

We establish a lower bound for the surface area of a closed, convex hypersurface in Euclidean space in terms of its displacement under continuous maps. As a result, a hypothesized lower bound for the volume of a Riemannian $n$-sphere,…

It is a well known phenomenon that many classical minimal surfaces in Euclidean space also exist with higher dihedral symmetry. More precisely, these surfaces are solutions to free boundary problems in a wedge bounded by two vertical planes…

Let $H_n$ be the minimal number of smaller homothetic copies of an $n$-dimensional convex body required to cover the whole body. Equivalently, $H_n$ can be defined via illumination of the boundary of a convex body by external light sources.…

Theoretical background is provided towards the mathematical foundation of the minimum enclosing ball problem. This problem concerns the determination of the unique spherical surface of smallest radius enclosing a given bounded set in the…

We show that the size of a minimal simplicial cover of a polytope $P$ is a lower bound for the size of a minimal triangulation of $P$, including ones with extra vertices. We then use this fact to study minimal triangulations of cubes, and…

In this paper we give a geometric argument for bounding the diameter of a connected compact surface (with boundary) of arbitrary codimension in Euclidean space in terms of Topping's diameter bound for closed surfaces (without boundary). The…

Motivated by the random covering problem and the study of Dirichlet uniform approximable numbers, we investigate the uniform random covering problem. Precisely, consider an i.i.d. sequence $\omega=(\omega_n)_{n\geq 1}$ uniformly distributed…

The isoperimetric problem asks for the maximum area of a region of given perimeter. It is natural to consider other measurements of a region, such as the diameter and width, and ask for the extreme value of one when another is fixed. The…

Determining the minimum density of a covering of $\mathbb{R}^{n}$ by Euclidean unit balls as $n\to\infty$ is a major open problem, with the best known results being the lower bound of $\left(\mathrm{e}^{-3/2}+o(1)\right)n$ by Coxeter, Few…

A small polygon is a polygon of unit diameter. The maximal area of a small polygon with $n=2m$ vertices is not known when $m\ge 7$. Finding the largest small $n$-gon for a given number $n\ge 3$ can be formulated as a nonconvex quadratically…

We obtain a new lower bound for the eternal vertex cover number of an arbitrary graph $G$, in terms of the cardinality of a vertex cover of minimum size in $G$ containing all its cut vertices. The consequences of the lower bound includes a…

Motivated by modern applications like image processing and wireless sensor networks, we consider a variation of the famous Kepler Conjecture. Given any infinite set of unit balls covering the whole space, we want to know the optimal (lim…

In this paper we present a new bound obtained with the probabilistic method for the solution of the Set Covering problem with unit costs. The bound is valid for problems of fixed dimension, thus extending previous similar asymptotic…

Using ideas from the geometry of compression, we improve on the current upper and lower bounds of the Heilbronn triangle problem. In particular, let $\Delta(s)$ denote the minimal area of the triangle induced by $s$ points on a unit disk.…

We give an overview of the 2023 Computational Geometry Challenge targeting the problem Minimum Coverage by Convex Polygons, which consists of covering a given polygonal region (possibly with holes) by a minimum number of convex subsets, a…

The width of a closed convex subset of Euclidean space is the distance between two parallel supporting planes. The Blaschke-Lebesgue problem consists of minimizing the volume in the class of convex sets of fixed constant width and is still…

We explore upper bounds on the covering radius of non-hollow lattice polytopes. In particular, we conjecture a general upper bound of $d/2$ in dimension $d$, achieved by the "standard terminal simplices" and direct sums of them. We prove…

In connection with an unsolved problem of Bang (1951) we give a lower bound for the sum of the base volumes of cylinders covering a d-dimensional convex body in terms of the relevant basic measures of the given convex body. As an…

Inspired by the boolean discrepancy problem, we study the following optimization problem which we term \textsc{Spherical Discrepancy}: given $m$ unit vectors $v_1, \dots, v_m$, find another unit vector $x$ that minimizes $\max_i \langle x,…