Analysis Seminar: A probabilistic approach to Lp affine isoperimetric inequalities

Abstract: In the class of convex sets, the isoperimetric inequality can be derived from several different affine inequalities. One example is
the Blaschke-Santalo inequality on the product of volumes of a convex body and its polar dual. Another example is the Busemann--Petty inequality for centroid bodies. In the 1990s, Lutwak and Zhang introduced a related functional analytic framework with their notion of Lp centroid bodies, for p>1. Lutwak raised the question of encompassing the non-convex star-shaped range when p<1 (including negative values). I will discuss a probabilistic approach to establishing isoperimetric inequalities in this range. It uses a new representation of star-shaped sets as special averages of convex sets, similar to harmonic analysis methods involving superposition of Gaussians. Based on joint work with R. Adamczak, G. Paouris, and P. Simanjuntak.

Host: Henri Martikainen