The Christoffel problem for the disk area measure

TL;DR

The paper resolves a concrete instance of the mixed Christoffel problem by taking the disk $\mathbb{D}$ as reference body and linking the disk area measure $S_1(K,\mathbb{D};\cdot)$ to the convex body $K$ via an integral representation that recovers the support function $h_K$ without assuming regularity. In the smooth regime, the problem becomes a linear differential equation on $\mathbb{S}^{n-1}$, and the authors derive a practical density condition ensuring both convexity and $C^2_+$ regularity of the solution. A key contribution is the a priori decomposition of $S_1(K,\mathbb{D},du)$ into a convolution over 2D projections, implemented through a disintegration over $E\in\mathrm{Gr}_2(\mathbb{R}^n,e_n)$, coupled with a 2D Christoffel problem to generate the needed data. The convexity criterion is then reduced to a Schur-complement positivity check on the Hessian, yielding necessary and sufficient conditions for a given density $q$ to arise from a $C^2_+$ disk-area measure, with uniqueness up to translation; in the zonal case, symmetry further simplifies the construction and yields immediate corollaries for revolution bodies.

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.

Theorems & Definitions (32)

• Theorem 1.1: Berg1969
• Theorem A
• Theorem B
• Lemma 2.1: Schneider2014*p. 215
• Theorem 2.2: Shenfeld2022
• Corollary 2.3
• Theorem 2.4: Ambrosio2008
• Corollary 2.5
• proof
• proof : Proof of \ref{['mthm:Christoffel_disk_gnrl']}