On some covering problems in geometry

We present a method to obtain upper bounds on covering numbers. As applications of this method, we reprove and generalize results of Rogers on economically covering Euclidean $n$-space with translates of a convex body, or more generally, any measurable set. We obtain a bound for the density of covering the $n$-sphere by rotated copies of a spherically convex set (or, any measurable set). Using the same method, we sharpen an estimate by Artstein--Avidan and Slomka on covering a bounded set by translates of another. The main novelty of our method is that it is not probabilistic. The key idea, which makes our proofs rather simple and uniform through different settings, is an algorithmic result of Lov\'asz and Stein.Comment: 9 pages. IMPORTANT CHANGE: In previous versions of the paper, the illumination problem was also considered, and I presented a construction of a body close to the Euclidean ball with high illumination number. Now, I removed this part from this manuscript and made it a separate paper, 'A Spiky Ball'. It can be found at http://arxiv.org/abs/1510.0078

Fractional illumination of convex bodies

We introduce a fractional version of the illumination problem of Gohberg, Markus, Boltyanski and Hadwiger, according to which every convex body in ${\mathbb R}^d$ is illuminated by at most $2^d$ directions. We say that a weighted set of points on ${\mathbb S}^{d-1}$ illuminates a convex body $K$ if for each boundary point of $K$, the total weight of those directions that illuminate $K$ at that point is at least one. We prove that the fractional illumination number of any o-symmetric convex body is at most $2^d$, and of a general convex body $\binom{2d}{d}$. As a corollary, we obtain that for any o-symmetric convex polytope with $k$ vertices, there is a direction that illuminates at least $\left\lceil\frac{k}{2^d}\right\rceil$ vertices

Ball and Spindle Convexity with respect to a Convex Body

Let $C\subset {\mathbb R}^n$ be a convex body. We introduce two notions of convexity associated to C. A set $K$ is $C$-ball convex if it is the intersection of translates of $C$, or it is either $\emptyset$, or ${\mathbb R}^n$. The $C$-ball convex hull of two points is called a $C$-spindle. $K$ is $C$-spindle convex if it contains the $C$-spindle of any pair of its points. We investigate how some fundamental properties of conventional convex sets can be adapted to $C$-spindle convex and $C$-ball convex sets. We study separation properties and Carath\'eodory numbers of these two convexity structures. We investigate the basic properties of arc-distance, a quantity defined by a centrally symmetric planar disc $C$, which is the length of an arc of a translate of $C$, measured in the $C$-norm, that connects two points. Then we characterize those $n$-dimensional convex bodies $C$ for which every $C$-ball convex set is the $C$-ball convex hull of finitely many points. Finally, we obtain a stability result concerning covering numbers of some $C$-ball convex sets, and diametrically maximal sets in $n$-dimensional Minkowski spaces.Comment: 27 pages, 5 figure

On multiple Borsuk numbers in normed spaces

Hujter and Lángi defined the k-fold Borsuk number of a set S in Euclidean n-space of diameter d > 0 as the smallest cardinality of a family F of subsets of S, of diameters strictly less than d, such that every point of S belongs to at least k members of F. We investigate whether a k-fold Borsuk covering of a set S in a �nite dimensional real normed space can be extended to a completion of S. Furthermore, we determine the k-fold Borsuk number of sets in not angled normed planes, and give a partial characterization for sets in angled planes