Characterization of self-polar convex functions

TL;DR

This paper characterizes all rotationally invariant self-polar convex functions on R^n, expanding the understanding of polarity beyond convex bodies to a broader class of convex functions.

Contribution

It provides a complete classification of rotationally invariant self-polar convex functions, a significant extension of polarity concepts from convex bodies to functions.

Findings

Only the Euclidean ball is self-polar among convex bodies.

Numerous self-polar convex functions exist beyond convex bodies.

Complete characterization of rotationally invariant self-polar convex functions.

Abstract

In a work by Artstein-Avidan and Milman the concept of polarity is generalized from the class of convex bodies to the larger class of convex functions. While the only self-polar convex body is the Euclidean ball, it turns out that there are numerous self-polar convex functions. In this work we give a complete characterization of all rotationally invariant self-polar convex functions on R^n.