Simplicial polytopes are a prominent class of objects of interest in many areas of mathematics. They are convex hulls of finite (sufficiently) general sets of points in Euclidean space. In 1981 Stanley finalized the proof of the so called g-theorem: an astonishing conjecture of McMullen about the combinatorial structure of such polytopes.

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

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