Mahler's conjecture in convex geometry: a summary and further numerical analysis

Abstract

In this thesis we study Mahler's conjecture in convex geometry, give a short summary about its history, gather and explain different approaches that have been used to attack the conjecture, deduce formulas to calculate the Mahler volume and perform numerical analysis on it. The conjecture states that the Mahler volume of any symmetric convex body, i.e. the product of the volume of the symmetric convex body and the volume of its dual body, is minimized by the (hyper-)cube. The conjecture was stated and solved in 1938 for the 2-dimensional case by Kurt Mahler. While the maximizer for this problem is known (it is the ball), the conjecture about the minimizer is still open for all dimensions greater than 2. A lot of effort has benn made to solve this conjecture, and many different ways to attack the conjecture, from simple geometric attempts to ones using sophisticated results from functional analysis, have all been tried unsuccesfully. We will present and discuss the most important approaches. Given the support function of the body, we will then introduce several formulas for the volume of the dual and the original body and hence for the Mahler volume. These formulas are tested for their effectiveness and used to perform numerical work on the conjecture. We examine the conjectured minimizers of the Mahler volume by approximating them in different ways. First the spherical harmonic expansion of their support functions is calculated and then the bodies are analyzed with respect to the length of that expansion. Afterwards the cube is further examined by approximating its principal radii of curvature functions, which involve Dirac delta functions.