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The Gauss image problem for pseudo-cones

Rolf Schneider

TL;DR

This work extends the Gauss image problem to $C$-pseudo-cones, asking when a finite measure $\mu$ on $\Omega_C$ can be realized as the Gauss image measure $\lambda(K,\cdot)$ of a $C$-pseudo-cone $K$ with respect to a fixed finite measure $\lambda$ on $\mathrm{cl}\,\Omega_{C^\circ}$. It develops a variational framework using Wulff shapes and convexifications, defining a functional $\Phi_{\mu,\lambda,\eta}$ whose minimizers yield a $C$-pseudo-cone $K$ satisfying $\mu=\lambda(K,\cdot)$ on expanding domains; the main result gives a sharp necessary-and-sufficient condition $\lambda(\Omega_{C^\circ})=\mu(\Omega_C)$ for existence. A uniqueness statement is proved under the assumption that $\lambda$ is positive on open sets and the pseudo-cones are restricted, showing that any two such $K$ with the same image measure must be dilates. By recasting the problem in the conic setting, the paper connects the Gauss image problem to variational methods for geometric measure problems and hints at links to optimal transport in cones.

Abstract

The Gauss image problem for convex bodies asks for the existence of a convex body that "links" two given measures on the unit sphere in a certain way. We treat here a corresponding question for pseudo-cones, that is, for unbounded closed convex sets strictly contained in their recession cones.

The Gauss image problem for pseudo-cones

TL;DR

This work extends the Gauss image problem to -pseudo-cones, asking when a finite measure on can be realized as the Gauss image measure of a -pseudo-cone with respect to a fixed finite measure on . It develops a variational framework using Wulff shapes and convexifications, defining a functional whose minimizers yield a -pseudo-cone satisfying on expanding domains; the main result gives a sharp necessary-and-sufficient condition for existence. A uniqueness statement is proved under the assumption that is positive on open sets and the pseudo-cones are restricted, showing that any two such with the same image measure must be dilates. By recasting the problem in the conic setting, the paper connects the Gauss image problem to variational methods for geometric measure problems and hints at links to optimal transport in cones.

Abstract

The Gauss image problem for convex bodies asks for the existence of a convex body that "links" two given measures on the unit sphere in a certain way. We treat here a corresponding question for pseudo-cones, that is, for unbounded closed convex sets strictly contained in their recession cones.

Paper Structure

This paper contains 6 sections, 14 theorems, 90 equations.

Table of Contents

  1. Introduction
  2. Fundamentals about pseudo-cones
  3. About $\lambda(K,\cdot)$
  4. Variational lemmas
  5. Proof of existence
  6. A uniqueness result

Key Result

Theorem 1

Let $\lambda$ be a measure on the Lebesgue measurable subsets of ${\rm cl}\,\Omega_{C^\circ}$ which is non-zero, finite, and zero on Borel sets of Hausdorff dimension $n-2$. Let $\mu$ be a Borel measure on $\Omega_C$. There exists a $C$-pseudo-cone $K$ with if and only if $\lambda(\Omega_{C^\circ})=\mu(\Omega_C)$.

Theorems & Definitions (26)

  • Theorem 1
  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Lemma 3
  • proof
  • Lemma 4
  • proof
  • Lemma 5
  • ...and 16 more