The Christoffel problem for the disk area measure

TL;DR

This paper addresses the mixed Christoffel problem for disk area measures, providing integral and differential equation formulations to reconstruct convex bodies from measures, with conditions ensuring convexity and regularity.

Contribution

It introduces an integral representation and a differential equation approach for the Christoffel problem involving disk area measures, extending previous results to less regular cases.

Findings

Derived an integral formula for support functions from disk area measures.

Reformulated the problem as a linear differential equation on the sphere.

Established conditions on measure density for convexity and regularity.

Abstract

The mixed Christoffel problem asks for necessary and sufficient conditions for a Borel measure on the Euclidean unit sphere to be the mixed area measure of some convex bodies, all but one of them are fixed. We consider the case in which the reference bodies are $(n-1)$-dimensional disks lying in a fixed hyperplane. We obtain an integral representation that reconstructs the support function of a convex body from its disk area measure, without any regularity assumptions. In the smooth setting, we reformulate the problem as a linear differential equation on the sphere, and derive a necessary and sufficient condition on the density of the disk area measure guaranteeing both convexity and regularity of the solution.