Isomorphic and almost-isometric problems in high-dimensional convex geometry

Abstract

The classical theorems of high-dimensional convex geometry exhibit a surprising level of regularity and order in arbitrary high-dimensional convex sets. These theorems are mainly concerned with the rough geometric features of general convex sets; the so-called “isomorphic” features. Recent results indicate that, perhaps, high-dimensional convex sets are also very regular on the almost-isometric scale. We review some related research directions in high-dimensional convex geometry, focusing in particular on the problem of geometric symmetrization.