Covering by homothets and illuminating convex bodies

The paper is devoted to coverings by translative homothets and illuminations of convex bodies. For a given positive number α and a convex body B, gα⁡(B) is the infimum of α-powers of finitely many homothety coefficients less than 1 such that there is a covering of B by translative homothets with these coefficients. hα⁡(B) is the minimal number of directions such that the boundary of B can be illuminated by this number of directions except for a subset whose Hausdorff dimension is less than α. In this paper, we prove that gα⁡(B)≤hα⁡(B), find upper and lower bounds for both numbers, and discuss several general conjectures. In particular, we show that hα⁡(B)\u3e2d−α for almost all α and d when B is the d-dimensional cube, thus disproving the conjecture from Brass, Moser, and Pach [Research problems in discrete geometry, Springer, New York, 2005]

Illuminating spindle convex bodies and minimizing the volume of spherical sets of constant width

A subset of the d-dimensional Euclidean space having nonempty interior is called a spindle convex body if it is the intersection of (finitely or infinitely many) congruent d-dimensional closed balls. The spindle convex body is called a "fat" one, if it contains the centers of its generating balls. The core part of this paper is an extension of Schramm's theorem and its proof on illuminating convex bodies of constant width to the family of "fat" spindle convex bodies.Comment: 17 page

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

On the X-ray number of almost smooth convex bodies and of convex bodies of constant width

The X-ray numbers of some classes of convex bodies are investigated. In particular, we give a proof of the X-ray Conjecture as well as of the Illumination Conjecture for almost smooth convex bodies of any dimension and for convex bodies of constant width of dimensions 3, 4, 5 and 6