The Centro-Affine Hadwiger Theorem

TL;DR

This paper classifies all upper semicontinuous, SL(n) invariant valuations on convex bodies containing the origin, showing they are linear combinations of fundamental geometric measures including Euler characteristic, volume, polar volume, and Orlicz surface areas.

Contribution

It provides a complete classification of certain valuations on convex bodies, extending the understanding of geometric invariants under SL(n) transformations.

Findings

Valuations are linear combinations of known geometric measures.

Includes the recently discovered Orlicz surface areas.

Classification applies to convex bodies containing the origin.

Abstract

All upper semicontinuous and SL(n) invariant valuations on convex bodies containing the origin in their interiors are completely classified. Each such valuation is shown to be a linear combination of the Euler characteristic, the volume, the volume of the polar body, and the recently discovered Orlicz surface areas.