Computer Science/Discrete Mathematics Seminar I

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.