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The Minkowski problem for the non-compact convex set with an asymptotic boundary condition

Ning Zhang

Abstract

In this paper, combining the covolume, we study the Minkowski theory for the non-compact convex set with an asymptotic boundary condition. In particular, the mixed covolume of two non-compact convex sets is introduced and its geometric interpretation is obtained by the Hadamard variational formula. The Brunn-Minkowski and Minkowski inequalities for covolume are established, and the equivalence of these two inequalities are discussed as well. The Minkowski problem for non-compact convex set is proposed and solved under the asymptotic conditions. In the end, we give a solution to the Minkowski problem for $σ$-finite measure on the conic domain $Ω_C$.

The Minkowski problem for the non-compact convex set with an asymptotic boundary condition

Abstract

In this paper, combining the covolume, we study the Minkowski theory for the non-compact convex set with an asymptotic boundary condition. In particular, the mixed covolume of two non-compact convex sets is introduced and its geometric interpretation is obtained by the Hadamard variational formula. The Brunn-Minkowski and Minkowski inequalities for covolume are established, and the equivalence of these two inequalities are discussed as well. The Minkowski problem for non-compact convex set is proposed and solved under the asymptotic conditions. In the end, we give a solution to the Minkowski problem for -finite measure on the conic domain .
Paper Structure (8 sections, 12 theorems, 138 equations)

This paper contains 8 sections, 12 theorems, 138 equations.

Table of Contents

  1. Introduction
  2. Background and Notations
  3. The asymptotic boundary set of non-compact convex sets
  4. Wulff shapes
  5. The Brunn-Minkowski inequality
  6. The related mixed volume
  7. The Minkowski problem for non-compact convex sets
  8. The Minkowski problem for C-coconvex sets

Key Result

Theorem 1.2

Let $K_0,K_1\in \mathcal{C}_{b}$ with $\mathsf{A}(K_0)=a\mathsf{A}(K_1)$ for some $a>0$, and let $\lambda\in (0,1)$. Then where $K_0^c\oplus K_1^c=\mathsf{A}(K_0+K_1)\backslash (K_0+K_1)$. Equality holds if and only if $K_0=aK_1$.

Theorems & Definitions (31)

  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Lemma 2.1
  • proof
  • Definition 2.4
  • ...and 21 more