MM-Estimates, Regular M-Ellipsoids and Distances Between Convex Bodies
Explore the geometric properties of convex bodies in n-dimensional Euclidean space through this lecture on MM*-estimates, regular M-ellipsoids, and distances between convex bodies. Delve into key concepts such as width, average width, and the product of mean widths (MM*) for convex bodies and their polars. Examine the differences between volume product and MM*, focusing on the latter's lack of affine invariance and its lower bound of 1. Investigate the minimization of MM* over linear or affine images of convex bodies, with particular attention to the distinction between origin-symmetric and non-symmetric cases. Learn about the relationship between mean width, MM*, and the efficiency of covering convex bodies with dilated Euclidean balls. Discover Rudelson's approach to bounding maximal distances between convex bodies in non-symmetric cases. Gain insights into seminal results in Asymptotic Geometric Analysis and explore the ongoing challenges in resolving the non-symmetric case.