Title: The affine surface area in Euclidean convex geometry and beyond
Speaker: Florian Besau (Postdoctoral Researcher, Goethe University Frankfurt)
Abstract: Loosely speaking we can describe Euclidean convex geometry as the study of “natural” functions on convex bodies (e.g. volume, perimeter, average width, etc.) that are invariant with respect to Euclidean motions, i.e., rotations and translations. In some cases these functions turn out to be invariant not only with respect to Euclidean motions of convex bodies, but also with respect to a larger group such as equi-affine or affine motions.
In the first part of this talk I will give a brief survey on the classification of some invariant functions, i.e. valuations, in Euclidean and affine convex geometry. We will see how the affine surface area appears naturally as equi-affine invariant valuation on convex bodies. In the second part I will focus on the connections between the affine surface area, the convex floating body and random polytopal approximation. In the final part I will talk about extensions of these notions and connections to a more general setting. These results answered many analogous questions raised in other spaces, such as the Euclidean unit sphere or hyperbolic space, and have only just become available through joint work with Monika Ludwig and Elisabeth Werner.