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Mixed Christoffel-Minkowski problems for bodies of revolution

Leo Brauner, Georg C. Hofstätter, Oscar Ortega-Moreno

TL;DR

This work resolves the mixed Christoffel-Minkowski problem for axially symmetric convex bodies by introducing a transformation framework that transfers measures from general revolution reference bodies to the disk. Central to the approach are the $T_R$-transform and the $\widehat{T}_{\mathcal{C}}$ dictionary, which yield a continuous transformation rule and sharp Firey-type boundary estimates, along with invertibility results that enable lifting disk solutions to arbitrary revolution references. The authors obtain a full characterization of measures that arise as $S_i(K,\mathcal{C};\cdot)$, establish Hadwiger-type theorems for $\mathrm{SO}(n-1)$-invariant valuations with restricted support, and extend the CM theory to anisotropic, axisymmetric settings with precise endpoint and pole conditions. Additionally, they provide a complete solution in the isotropic disk case and a robust transfer mechanism to general reference bodies, clarifying uniqueness and degeneracy phenomena and enriching the understanding of local and boundary behavior of mixed area measures in revolution geometry.

Abstract

The mixed Christoffel-Minkowski 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, one of which, appearing multiple times, is free and the rest are fixed. In the case where all bodies involved are symmetric around a common axis, we provide a complete solution to this problem, without assuming any regularity. In particular, we refine Firey's classification of area measures of figures of revolution. In our argument, we introduce an easy way to transform mixed area measures and mixed volumes involving axially symmetric bodies, and we significantly improve Firey's estimate on the local behavior of area measures. As a secondary result, we obtain a family of Hadwiger type theorems for convex valuations that are invariant under rotations around an axis.

Mixed Christoffel-Minkowski problems for bodies of revolution

TL;DR

This work resolves the mixed Christoffel-Minkowski problem for axially symmetric convex bodies by introducing a transformation framework that transfers measures from general revolution reference bodies to the disk. Central to the approach are the -transform and the dictionary, which yield a continuous transformation rule and sharp Firey-type boundary estimates, along with invertibility results that enable lifting disk solutions to arbitrary revolution references. The authors obtain a full characterization of measures that arise as , establish Hadwiger-type theorems for -invariant valuations with restricted support, and extend the CM theory to anisotropic, axisymmetric settings with precise endpoint and pole conditions. Additionally, they provide a complete solution in the isotropic disk case and a robust transfer mechanism to general reference bodies, clarifying uniqueness and degeneracy phenomena and enriching the understanding of local and boundary behavior of mixed area measures in revolution geometry.

Abstract

The mixed Christoffel-Minkowski 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, one of which, appearing multiple times, is free and the rest are fixed. In the case where all bodies involved are symmetric around a common axis, we provide a complete solution to this problem, without assuming any regularity. In particular, we refine Firey's classification of area measures of figures of revolution. In our argument, we introduce an easy way to transform mixed area measures and mixed volumes involving axially symmetric bodies, and we significantly improve Firey's estimate on the local behavior of area measures. As a secondary result, we obtain a family of Hadwiger type theorems for convex valuations that are invariant under rotations around an axis.

Paper Structure

This paper contains 23 sections, 54 theorems, 198 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Main result
  3. Ideas of the proof
  4. A family of Hadwiger type theorems
  5. Plan of the article
  6. Preliminaries
  7. Convex geometry
  8. Valuations
  9. Functions of locally bounded variation
  10. Preliminaries on bodies of revolution
  11. Mixed spherical projections and liftings
  12. Transforming mixed volumes
  13. Definition of the T R transform and the semigroup property
  14. The continuous transformation rule
  15. Local behavior of mixed area measures
  16. ...and 8 more sections

Key Result

Theorem 1.1

Let $1\leq i < n-1$ and $\mu$ be a centered, zonal measure on $\mathbb{S}^{n-1}$ with a strictly positive density $q \in C(\mathbb{S}^{n-1})$. Then there exists a body of revolution $K\in\mathcal{K}(\mathbb{R}^n)$ with $\mu=S_i(K,{}\cdot{})$ if and only if for all $t\in (-1,1)$,

Figures (1)

  • Figure :

Theorems & Definitions (104)

  • Theorem 1.1: Firey1970
  • Theorem A
  • Corollary A
  • Theorem B
  • Theorem C
  • Theorem D
  • Theorem E
  • Lemma 2.1: Schneider2014*p. 215
  • Definition 2.2: Knoerr2021
  • Theorem 2.3: Brauner2024a*Cor. C
  • ...and 94 more