Related papers: An Upper Bound for Lebesgue's Covering Problem
Given $N$ geodesic caps on the unit sphere in $\mathbb{R}^d$, and whose total normalized surface area sums to one, what is the maximal surface area their union can cover? In this work, we provide an asymptotically sharp upper bound for an…
Let $C_{\theta}$ be a circular cone in Euclidean space $\mathbb{R}^{3}$,which apex is the origin and apex angle of the cone is $\theta\in \left(\pi/3, \pi\right)$. Let $M_\theta$ be the class of compact convex domains in Euclidean space…
We consider a class of convex optimization problems over the simplex of probability measures. Our framework comprises optimal experimental design (OED) problems, in which the measure over the design space indicates which experiments are…
We provide the currently fastest randomized (1+epsilon)-approximation algorithm for the closest vector problem in the infinity norm. The running time of our method depends on the dimension n and the approximation guarantee epsilon by 2^O(n)…
Suppose that each proper subset of a set $S$ of points in a vector space is contained in the union of planes of specified dimensions, but $S$ itself is not contained in any such union. How large can $|S|$ be? We prove a general upper bound…
The Illumination Problem may be phrased as the problem of covering a convex body in Euclidean $n$-space by a minimum number of translates of its interior. By a probabilistic argument, we show that, arbitrarily close to the Euclidean ball,…
The covering radius problem is a question in coding theory concerned with finding the minimum radius $r$ such that, given a code that is a subset of an underlying metric space, balls of radius $r$ over its code words cover the entire metric…
An essential cover of the vertices of the $n$-cube $\{0,1\}^n$ by hyperplanes is a minimal covering where no hyperplane is redundant and every variable appears in the equation of at least one hyperplane. Linial and Radhakrishnan gave a…
We study the problem of finding, for a given one-dimensional topological space $X$, a cover of $X$ of smallest size by geodesics with respect to some metric. The infimal size of such a set is called the metric geodesic cover number of $X$.…
We show that a realization of a closed connected PL-manifold of dimension n-1 in n-dimensional Euclidean space (n>2) is the boundary of a convex polyhedron (finite or infinite) if and only if the interior of each (n-3)-face has a point,…
In this paper we study some cube packing problems. In particular we are interested in compact subsets of $\mathbb{R}^n,n\geq 2$, which contain boundaries of cubes with all side lengths in $(0,1)$. We show here that such sets must have lower…
We consider the problem of minimizing a continuous function f over a compact set K. We analyze a hierarchy of upper bounds proposed by Lasserre in [SIAM J. Optim. 21(3) (2011), pp. 864--885], obtained by searching for an optimal probability…
The Sphere Covering Inequality was introduced in \cite{GM} (\emph{Invent. Math.}, 2018) as a sharp geometric inequality that provides a lower bound for the total area of two distinct surfaces of Gaussian curvature 1. These surfaces are…
We present filling as a type of spatial subdivision problem similar to covering and packing. Filling addresses the optimal placement of overlapping objects lying entirely inside an arbitrary shape so as to cover the most interior volume. In…
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…
We study a specific convex maximization problem in n-dimensional space. The conjectured solution is proved to be a vertex of the polyhedral feasible region, but only a partial proof of local maximality is known. Integer sequences with…
We prove that the first gap of $\mathbb R$-complementary thresholds of surfaces is $\frac{1}{13}$. More precisely, the largest $\mathbb R$-complementary threshold for surfaces that is strictly less than $1$ is $\frac{12}{13}$. This result…
We consider the set multi-cover problem in geometric settings. Given a set of points P and a collection of geometric shapes (or sets) F, we wish to find a minimum cardinality subset of F such that each point p in P is covered by (contained…
Coverings of convex bodies have emerged as a central component in the design of efficient solutions to approximation problems involving convex bodies. Intuitively, given a convex body $K$ and $\epsilon> 0$, a covering is a collection of…
Consider a set P of points in the unit square U, one of them being the origin. For each point p in P you may draw a rectangle in U with its lower-left corner in p. What is the maximum area such rectangles can cover without overlapping each…