The Christoffel problem for the disk area measure

2025/08/13 by Brauner, Leo, Hofstätter, Georg C., Ortega-Moreno, Oscar · 4 citations
#52A20 #52A39 #Analysis of PDEs (math.AP) #Differential Geometry (math.DG) #FOS: Mathematics #Metric Geometry (math.MG)

paper · doi:10.48550/arxiv.2508.09800

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 provide a solution in the case where the fixed reference bodies are (n-1)-dimensional disks with parallel axes and the measure has no mass at the poles of the sphere determined by this axis.

Citations

• A constant rank theorem for linear elliptic equations on the sphere with applications to the mixed Christoffel problem
• Explicit solutions to Christoffel-Minkowski problems and Hessian equations under rotational symmetries
• The Klain approach to zonal valuations
• Kubota-type formulas and supports of mixed measures
• The Christoffel problem by fundamental solution of the Laplace equation
• Spherical Harmonics

Cited by

• A constant rank theorem for linear elliptic equations on the sphere with applications to the mixed Christoffel problem
• Anisotropic fractional area measures
• Explicit solutions to Christoffel-Minkowski problems and Hessian equations under rotational symmetries
• Mixed Christoffel-Minkowski problems for bodies of revolution

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• Mixed Christoffel-Minkowski problems for bodies of revolution
• The capillary Christoffel-Minkowski problem
• Explicit solutions to Christoffel-Minkowski problems and Hessian equations under rotational symmetries
• Lorentzian area measures and the Christoffel problem
• The Christoffel problem and two analogs of the Minkowski problem in Riemannian space