Centro-Affine Invariants for Smooth Convex Bodies

TL;DR

This paper introduces new centro-affine differential invariants for smooth convex bodies using a flow approach, establishing sharp inequalities and providing a geometric interpretation for recent affine surface area concepts.

Contribution

It develops a novel centro-affine flow method to generate invariants, derives new inequalities, and offers a geometric understanding of recent affine surface area.

Findings

New centro-affine invariants established

Sharp isoperimetric inequalities proved

Geometric interpretation of recent affine surface area provided

Abstract

Employing a centro-affine flow on smooth convex bodies, we generate new centro-affine differential invariants. One class of the newly defined invariants is the object of a sharp isoperimetric inequality, while other new inequalities on known centro-affine invariants are obtained as a byproduct of the flow's study. Furthermore, this approach led to a geometric interpretation of a new affine surface area recently introduced by Ludwig and Reitzner.