Title:  Polar Self-Dual Polytopes and Mahler Volume

Abstract:  The dual of a polytope is a well known topic in polyhedral geometry. There are several notions of what it means for a polytope to be self-dual. Perhaps the most common idea of self-dual would be that a polytope is combinatorially equivalent to its dual (face poset isomorphism). In Stephen's recent work it has been helpful to consider the strictest form of self-duality called polar self-dual, where we require that an embedded polytope be exactly equal to its dual as a subset of real space. We will consider some of the results and conjectures relating to polar self-dual polytopes.

Another famous problem from the theory of convex duality is finding the extremizers of the Mahler volume. This is defined as the product of the area of the convex figure and the area of its dual. Using Steiner symmetrization, we will show the maximizer of this volume is the circle.