Floating Bodies and Approximation - Part 2

Presented on December 11, 2019 at 9:10 a.m. in the Bill Moore Student Success Center, Press Rooms A & B, Georgia Tech.Workshop in Convexity and Geometric Aspects of Harmonic AnalysisElisabeth Werner, Case Western Reserve University, Department of Mathematics, Applied Mathematics and StatisticsRuntime: 54:55 minutesTwo important closely related notions in affine convex geometry are the floating body and the affine surface area of a convex body. The floating body of a convex body is obtained by cutting off caps of volume less or equal to a fixed positive constant. Taking the right-derivative of the volume of the floating body gives rise to an affine invariant, the affine surface area. This was established for all convex bodies in all dimensions by Schuett and Werner. There is a natural inequality associated with affine surface area, the affine isoperimetric inequality, which states that among all convex bodies, with fixed volume, affine surface area is maximized for ellipsoids. Due to its important properties, which make them effective and powerful tools, affine surface area and floating body are omnipresent in geometry and have applications in many other areas of mathematics, e.g., in problems of approximation of convex bodies by polytopes and for the notion of halfspace depth for multivariate data from statistics