A Geometric Lower Bound Theorem

TL;DR

This paper proves a conjecture linking convex body approximation by polytopes to topological face numbers, using advanced chordality concepts, and provides asymptotically tight bounds on these face numbers for smooth convex bodies.

Contribution

It resolves Kalai's conjecture connecting approximation theory, face numbers, and Betti numbers, introducing higher chordality notions and bounds for smooth convex bodies.

Findings

Proves Kalai's conjecture on convex body approximation

Establishes lower bounds on g-numbers for C^2-convex bodies

Uses higher chordality concepts in the proof

Abstract

We resolve a conjecture of Kalai relating approximation theory of convex bodies by simplicial polytopes to the face numbers and primitive Betti numbers of these polytopes and their toric varieties. The proof uses higher notions of chordality. Further, for C^2-convex bodies, asymptotically tight lower bounds on the g-numbers of the approximating polytopes are given, in terms of their Hausdorff distance from the convex body.