# Fractional covering numbers, with an application to the Levi-Hadwiger conjecture for convex bodies

 Computer Science/Discrete Mathematics Seminar I Topic: Fractional covering numbers, with an application to the Levi-Hadwiger conjecture for convex bodies Speaker: Boaz Slomka Affiliation: Tel Aviv University; Member, School of Mathematics Date: Monday, October 21 Time/Room: 11:15am - 12:15pm/S-101

Let $K$ and $T$ be convex bodies in the $n$-dimensional Euclidean space. The covering number of $K$ by $T$ is the minimal number of translates of $T$ required to cover $K$ entirely. One open question regarding this classical notion is the Levi-Hadwiger conjecture which states that every $n$-dimensional convex body can be covered by $2^n$ slightly smaller homothetic copies of itself. We will discuss the notion of fractional covering numbers and some inequalities comparing classical covering numbers with fractional ones. As a consequence, we give a new proof for Rogers' bound for the covering number of a convex body by smaller homothetic copies of itself. Based on a joint work with Shiri Artstein-Avidan.