Affine invariant maps for log-concave functions

TL;DR

This paper extends the concept of affine invariant points and maps from convex sets to log-concave functions, introducing new invariants based on symmetry and classical centers like the Santaló point.

Contribution

It generalizes affine invariance notions to functions, providing new examples of invariant points and maps such as floating, John, and Löwner functions for log-concave functions.

Findings

Classical centers like the Santaló point are affine invariant for log-concave functions.

Floating, John, and Löwner functions serve as new affine invariant maps.

These invariants reveal symmetry properties of log-concave functions.

Abstract

Affine invariant points and maps for sets were introduced by Gr\"unbaum to study the symmetry structure of convex sets. We extend these notions to a functional setting. The role of symmetry of the set is now taken by evenness of the function. We show that among the examples for affine invariant points are the classical center of gravity of a log-concave function and its Santal\'o point. We also show that the recently introduced floating functions and the John- and L\"owner functions are examples of affine invariant maps. Their centers provide new examples of affine invariant points for log-concave functions.