The Christoffel problem for the disk area measure

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.

💡 Research Summary

The paper addresses a specific instance of the mixed Christoffel problem: given a Borel measure on the unit sphere, determine when it can be realized as the mixed area measure S₁(K, D,·) of a convex body K with respect to a fixed reference body D, where D is the (n‑1)-dimensional unit disk orthogonal to the nth coordinate axis. The authors provide two complementary solutions—one that works without any smoothness assumptions on K, and another that yields explicit, verifiable conditions when K is of class C²⁺ (i.e., its boundary is a C² hypersurface with everywhere positive Gauss curvature).

Background and Motivation
The classical Christoffel problem asks for necessary and sufficient conditions for a measure μ on Sⁿ⁻¹ to be the first-order area measure S₁(K,·) of some convex body K. Berg and Firey solved this by expressing μ as the image of a Green’s function applied to the support function h_K, leading to the condition that a certain integral (involving the kernel g_n) must itself be a support function. In the mixed setting, one replaces the unit ball in the definition of S₁ by an arbitrary convex body C, obtaining mixed area measures S₁(K, C;·). While general sufficient conditions have been obtained (e.g., by Colesanti–Focardi–Guan–Salani), a complete necessary and sufficient characterization is known only for C = Bⁿ (the original Christoffel problem).

Main Contributions

1. Theorem A – Integral Representation without Regularity

   • Assumptions: μ is a finite, centered Borel measure on Sⁿ⁻¹ with no mass at the poles ±e_n. Its push‑forward under the map π: u ↦ span{e_n, u} (which sends a direction to the unique 2‑plane containing e_n) is absolutely continuous with a continuous density.
   • Disintegration: Using the measure‑theoretic disintegration theorem, μ can be written as an integral over the Grassmannian Gr₂(ℝⁿ, e_n) of measures μ_E supported on the great circles S¹(E).
   • 2‑Dimensional Christoffel Problem: For each plane E, the classical 2‑dimensional Christoffel problem is solved via Berg’s Green function g₂(t)=√{1‑t²}(π‑arccos t)+c_n t, yielding a support function h_{K|E}.
   • Compatibility and Convexity: The collection {h_{K|E}} must satisfy a global integral equation (1.6) that guarantees they glue together to a single support function h_K on Sⁿ⁻¹. This condition encodes both the centering of μ_E (zero first moment) and the convexity of the resulting body.
   • Result: μ = S₁(K, D,·) for some convex K iff (i) π_*μ has a continuous density, (ii) each μ_E is centered, and (iii) the integral equation (1.6) holds. The body K is unique up to translation.

2. Theorem B – Differential Equation Approach for C²⁺ Bodies

   • Setup: Assume K ∈ K(ℝⁿ) is C²⁺, so its support function h_K ∈ C²(Sⁿ⁻¹). Parameterize Sⁿ⁻¹ by polar coordinates (θ, w) with θ∈