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Radial measures of pseudo-cones

Rolf Schneider

TL;DR

This work extends Minkowski-type theory to $C$-pseudo-cones by developing a radial-analytic framework based on the $F$-radial measure $\mu_{F,K}$ and the radial Gauss map $\alpha_K$. It proves a key derivative formula linking Wulff-perturbed pseudo-cones to $\mu_{F,K}$ (Theorem $T2.1$) and specializes to negative exponents to connect with dual curvature measures $\widetilde{C}_q(K,\cdot)$ and dual volumes $\widetilde{V}_q(K)$. The main contribution is solving the dual Minkowski problem for negative exponents: for any nonzero finite Borel measure $\varphi$ on ${\rm cl}\,\Omega_{C^\circ}$ and $q<0$, there exists $K\in ps(C)$ with $\widetilde{C}_q(K,\cdot)=\varphi$, established via a variational approach; uniqueness remains open. Overall, the paper broadens Minkowski-type results to pseudo-cones and provides a complete negative-exponent solvability theory, enriching the interplay between radial measures and convex-analytic perturbations.

Abstract

We consider $C$-pseudo-cones, that is, closed convex sets $K \subset{\mathbb R}^n$ with $o\notin K\subset C$, for which $C$ is the recession cone. Here $C$ is a given closed convex cone in ${\mathbb R}^n$, pointed and with nonempty interior. We define a class of measures for such pseudo-cones and show how they can be interpreted as derivative measures. For a subclass of these measures, namely for dual curvature measures with negative indices, we solve a Minkowski type existence problem.

Radial measures of pseudo-cones

TL;DR

This work extends Minkowski-type theory to -pseudo-cones by developing a radial-analytic framework based on the -radial measure and the radial Gauss map . It proves a key derivative formula linking Wulff-perturbed pseudo-cones to (Theorem ) and specializes to negative exponents to connect with dual curvature measures and dual volumes . The main contribution is solving the dual Minkowski problem for negative exponents: for any nonzero finite Borel measure on and , there exists with , established via a variational approach; uniqueness remains open. Overall, the paper broadens Minkowski-type results to pseudo-cones and provides a complete negative-exponent solvability theory, enriching the interplay between radial measures and convex-analytic perturbations.

Abstract

We consider -pseudo-cones, that is, closed convex sets with , for which is the recession cone. Here is a given closed convex cone in , pointed and with nonempty interior. We define a class of measures for such pseudo-cones and show how they can be interpreted as derivative measures. For a subclass of these measures, namely for dual curvature measures with negative indices, we solve a Minkowski type existence problem.
Paper Structure (4 sections, 4 theorems, 60 equations)

This paper contains 4 sections, 4 theorems, 60 equations.

Table of Contents

  1. Introduction
  2. An analogue for pseudo-cones
  3. Proof of Theorem \ref{['T2.1']}
  4. The dual Minkowski problem for negative exponents

Key Result

Theorem 1

Let $F$ and $G$ be as defined above. Let $K\in ps(C)$. Let $g:{\rm cl}\,\Omega_{C^\circ}\to{\mathbb R}$ be continuous. Let $K_t$ be the Wulff shape associated with $(C, \overline h_Ke^{tg})$ for $|t|\le 1$ (say). Then

Theorems & Definitions (7)

  • Theorem 1
  • Lemma 1
  • proof
  • proof
  • Theorem 2
  • Lemma 2
  • proof