Affine Invariant Maps for Log-Concave Functions

Abstract

Affine invariant points and maps for sets were introduced by Grünbaum 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 Santaló point. We also show that the recently introduced floating functions and the John- and Löwner functions are examples of affine invariant maps. Their centers provide new examples of affine invariant points for log-concave functions.