Title: Fractional covering numbers with an application to the Levi-Hadwiger
problem
Abstract: 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 too
cover K entirely .
One open problem 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. The conjecture also states that this bound
is optimal with equality only for parallelotopes.
We will discuss the notions of fractional covering and separation numbers and
show that there is a strong duality relation between them. We will formulate the
fractional version of the Levi-Hadwiger problem, and prove it for centrally
symmetric convex bodies (including the equality case). We will also discuss some
inequalities comparing classical covering numbers with fractional ones. As a
consequence, we will give a new proof for Roger's bound for the classical
Levi-Hadwiger problem.
Based on a joint work with Shiri Artstein-Avidan