Improved bounds for Hadwiger's covering problem via thin-shell estimates

A central problem in discrete geometry, known as Hadwiger's covering problem, asks what the smallest natural number $N\left(n\right)$ is such that every convex body in ${\mathbb R}^{n}$ can be covered by a union of the interiors of at most $N\left(n\right)$ of its translates. Despite continuous efforts, the best general upper bound known for this number remains as it was more than sixty years ago, of the order of ${2n \choose n}n\ln n$. In this note, we improve this bound by a sub-exponential factor. That is, we prove a bound of the order of ${2n \choose n}e^{-c\sqrt{n}}$ for some universal constant $c>0$. Our approach combines ideas from previous work by Artstein-Avidan and the second named author with tools from Asymptotic Geometric Analysis. One of the key steps is proving a new lower bound for the maximum volume of the intersection of a convex body $K$ with a translate of $-K$; in fact, we get the same lower bound for the volume of the intersection of $K$ and $-K$ when they both have barycenter at the origin. To do so, we make use of measure concentration, and in particular of thin-shell estimates for isotropic log-concave measures. Using the same ideas, we establish an exponentially better bound for $N\left(n\right)$ when restricting our attention to convex bodies that are $\psi_{2}$. By a slightly different approach, an exponential improvement is established also for classes of convex bodies with positive modulus of convexity

The unrestricted blocking number in convex geometry

Let K be a convex body in \mathbb{R}^n. We say that a set of translates \left \{ K + \underline{u}_i \right \}_{i=1}^{p} block K if any other translate of K which touches K, overlaps one of K + \underline{u}_i, i = 1, . . . , p. The smallest number of non-overlapping translates (i.e. whose interiors are disjoint) of K, all of which touch K at its boundary and which block any other translate of K from touching K is called the Blocking Number of K and denote it by B(K). This thesis explores the properties of the blocking number in general but the main purpose is to study the unrestricted blocking number B_\alpha(K), i.e., when K is blocked by translates of \alpha K, where \alpha is a fixed positive number and when the restrictions that the translates are non-overlapping or touch K are removed. We call this number the Unrestricted Blocking Number and denote it by B_\alpha(K). The original motivation for blocking number is the following famous problem: Can a rigid material sphere be brought into contact with 13 other such spheres of the same size? This problem was posed by Kepler in 1611. Although this problem was raised by Kepler, it is named after Newton since Newton and Gregory had a dispute over the solution which was eventually settled in Newton’s favour. It is called the Newton Number, N(K) of K and is defined to be the maximum number of non-overlapping translates of K which can touch K at its boundary. The well-known dispute between Sir Isaac Newton and David Gregory concerning this problem, which Newton conjectured to be 12, and Gregory thought to be 13, was ended 180 years later. In 1874, the problem was solved by Hoppe in favour of Newton, i.e., N(\beta^3) = 12. In his proof, the arrangement of 12 unit balls is not unique. This is thought to explain why the problem took 180 years to solve although it is a very natural and a very simple sounding problem. As a generalization of the Newton Number to other convex bodies the blocking number was introduced by C. Zong in 1993. “Another characteristic of mathematical thought is that it can have no success where it cannot generalize.” C. S. Pierce As quoted above, in mathematics generalizations play a very important part. In this thesis we generalize the blocking number to the Unrestricted Blocking Number. Furthermore; we also define the Blocking Number with negative copies and denote it by B_(K). The blocking number not only gives rise to a wide variety of generalizations but also it has interesting observations in nature. For instance, there is a direct relation to the distribution of holes on the surface of pollen grains with the unrestricted blocking number

Asymptotics of generalized Hadwiger numbers

We give asymptotic estimates for the number of non-overlapping homothetic copies of some centrally symmetric oval $B$ which have a common point with a 2-dimensional domain $F$ having rectifiable boundary, extending previous work of the L.Fejes-Toth, K.Borockzy Jr., D.G.Larman, S.Sezgin, C.Zong and the authors. The asymptotics compute the length of the boundary $\partial F$ in the Minkowski metric determined by $B$. The core of the proof consists of a method for sliding convex beads along curves with positive reach in the Minkowski plane. We also prove that level sets are rectifiable subsets, extending a theorem of Erd\"os, Oleksiv and Pesin for the Euclidean space to the Minkowski space.Comment: 20p, 9 figure