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The growth rate of surface area measure for noncompact convex sets with prescribed asymptotic cone

Vadim Semenov, Yiming Zhao

Abstract

The Minkowski problem for a class of unbounded closed convex sets is considered. This is equivalent to a Monge-Ampère equation on a bounded convex open domain with possibly non-integrable given data. A complete solution (necessary and sufficient condition for existence and uniqueness) in dimension 2 is presented. In higher dimensions, partial results are demonstrated.

The growth rate of surface area measure for noncompact convex sets with prescribed asymptotic cone

Abstract

The Minkowski problem for a class of unbounded closed convex sets is considered. This is equivalent to a Monge-Ampère equation on a bounded convex open domain with possibly non-integrable given data. A complete solution (necessary and sufficient condition for existence and uniqueness) in dimension 2 is presented. In higher dimensions, partial results are demonstrated.
Paper Structure (8 sections, 38 theorems, 175 equations)

This paper contains 8 sections, 38 theorems, 175 equations.

Table of Contents

  1. Introduction
  2. Pseudo cones and $C$-asymptotic sets
  3. Some results from PDE
  4. The Minkowski problem for $C$-asymptotic sets in dimension 2
  5. Necessity of the condition
  6. Sufficiency of the condition
  7. Higher Dimensions
  8. A nice $C$ where global integrability condition is not enough

Key Result

Theorem 1.2

Let $C\subset \mathbb{R}^2$ be a pointed, closed, convex cone with a nonempty interior and $\mu$ be a Borel measure on $\Omega$. There exists a unique $C$-asymptotic set $K\subset C$ such that if and only if

Theorems & Definitions (69)

  • Theorem 1.2
  • Lemma 2.1
  • Corollary 2.2
  • Definition 2.3
  • Lemma 2.4
  • Lemma 2.5
  • proof
  • Theorem 2.6
  • Proposition 2.7
  • proof
  • ...and 59 more