Department of
Applied
Mathematics

Floating bodies and approximation of convex bodies by polytopes

Elisabeth Werner

Abstract

How well can a convex body be approximated by a polytope? This is a fundamental question in convex geometry, also in view of applications in many other areas of mathematics and related fields. It often involves side conditions like a prescribed number of vertices and a requirement that the body contains the polytope or vice versa. Accuracy of approximation is often measured in the symmetric difference metric but other metrics can and have been considered. We will present several results, mostly related to approximation by "random polytopes". We will introduce floating bodies and an affine invariant from affine differential geometry associated to them, the affine surface area. This affine invariant appears naturally as an important ingredient in approximation questions.