On the functional Minkowski problem

TL;DR

This paper fully solves the functional Minkowski problem by characterizing which measure pairs arise from log-concave functions, establishing their continuity, and introducing a new radial function concept for convex functions.

Contribution

It provides a complete solution to the functional Minkowski problem, including measure characterization, continuity results, and a novel radial function construction.

Findings

Characterization of measure pairs as surface area measures of log-concave functions

Continuity of surface area measures under function convergence

Continuity of solutions with respect to measure data

Abstract

To every log-concave function $f$ one may associate a pair of measures $(\mu_{f},\nu_{f})$ which are the surface area measures of $f$. These are a functional extension of the classical surface area measure of a convex body, and measure how the integral $\int f$ changes under perturbations. The functional Minkowski problem then asks which pairs of measures can be obtained as the surface area measures of a log-concave function. In this work we fully solve this problem. Furthermore, we prove that the surface area measures are continuous in correct topology: If $f_{k}\to f$, then $\left(\mu_{f_{k}},\nu_{f_{k}}\right)\to\left(\mu_{f},\nu_{f}\right)$ in the appropriate sense. Finding the appropriate mode of convergence of the pairs $\left(\mu_{f_{k}},\nu_{f_{k}}\right)$ sheds a new light on the construction of functional surface area measures. To prove this continuity theorem we associate to every convex function a new type of radial function, which seems to be an interesting construction on its own right. Finally, we prove that the solution to functional Minkowski problem is continuous in the data, in the sense that if $\left(\mu_{f_{k}},\nu_{f_{k}}\right)\to\left(\mu_{f},\nu_{f}\right)$ then $f_{k}\to f$ up to translations.