Analysis of polarity

TL;DR

This paper introduces a differential theory for the polarity transform applicable to geometric convex functions, enabling solutions to certain first and second order equations and a new method for interpolating convex functions.

Contribution

It develops a novel differential framework for the polarity transform for geometric convex functions, extending analytical tools and interpolation methods.

Findings

Provides a differential theory for the polarity transform.

Solves first order Hamilton-Jacobi and conservation law equations.

Introduces a canonical interpolation method for convex functions.

Abstract

We develop a differential theory for the polarity transform parallel to that for the Legendre transform, which is applicable when the functions studied are "geometric convex", namely convex, non-negative and vanish at the origin. This analysis may be used to solve a family of first order equations reminiscent of Hamilton--Jacobi and conservation law equations, as well as some second order Monge-Ampere type equations. A special case of the latter, that we refer to as the homogeneous polar Monge--Ampere equation, gives rise to a canonical method of interpolating between convex functions.