Explore the intricacies of proving isoperimetric inequalities through random sampling in this illuminating lecture by Peter Pivovarov. Delve into the world of convex sets and affine inequalities, examining fundamental constructions like polar bodies and centroid bodies. Discover how these concepts relate to strengthened isoperimetric theorems proven by Blaschke, Busemann, and Petty. Learn about the analytic framework developed by Lutwak, Yang, and Zhang for Lp affine isoperimetric inequalities. Investigate the challenges of establishing isoperimetric inequalities for non-convex Lp objects when p is less than 1 or negative, and understand the importance of this range in bridging Brunn-Minkowski theory and dual Brunn-Minkowski theory. Explore a probabilistic approach to proving Lp affine isoperimetric inequalities in the non-convex range, drawing inspiration from geometric probability concepts like Sylvester's four-point problem. Gain insights into empirical definitions of polar bodies and their Lp-analogues, and understand how these empirical versions provide a bridge between convex and non-convex worlds. This lecture, supported by NSF Grant DMS-2105468, is based on joint work with R. Adamczak, G. Paouris, and P. Simanjuntak. Read more