Jose ́ Casas Samper

University of Washington
, BH 401

Abstract

Combinatorics of polytopes and approximations of smoothconvex bodies

Simplicial polytopes are a prominent class of objects of interest inmany areas of mathematics. They are convex hulls of finite (sufficiently) generalsets of points in Euclidean space. In 1981 Stanley finalized the proof of the socalledg-theorem: an astonishing conjecture of McMullen about the combinatorialstructure of such polytopes.

Another mainstream field of research in which simplicial polytopes play a promi-nent role is the theory of convex bodies and metric structures on the spaces of suchbodies. Polytopes are dense in almost every reasonable metric for the space con-vex bodies ofRd. A question that has been widely studied is the following. Fix aconvex body and a sequence of polytopes that approximates it. Which restrictionsare imposed on the combinatorial data of the polytopes as the quality of approxi-mation increases. B ̈or ̈oczky, building on work of Schneider, has a series of resultsthat characterize many invariants of such polytopes. However, the combinatorialdata that is understood with these classical techniques is way more limited thanthe combinatorial data that appears in the g-theorem.

The goal of this talk is to introduce both theories and explain a new connectionbetween them. In particular, we present a solution to conjecture of Kalai relatingthe g-theorem with approximating sequences to smooth convex bodies and explainhow it vastly generalizes the theorems of B ̈or ̈oczky while strengthening the g-theorem to take into consideration more geometry of the polytopes. No previousknowledge about polytopes will be assumed. The talk is based on joint work withKarim Adiprasito and Eran Nevo.