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The Logarithmic Minkowski conjecture and the $L_p$-Minkowski Problem

Karoly J. Boroczky

Abstract

The current state of art concerning the $L_p$ Minkowski problem as a Monge-Ampere equation on the sphere and Lutwak's Logarithmic Minkowski conjecture about the uniqueness of even solution in the $p=0$ case are surveyed and connections to many related problems are discussed.

The Logarithmic Minkowski conjecture and the $L_p$-Minkowski Problem

Abstract

The current state of art concerning the Minkowski problem as a Monge-Ampere equation on the sphere and Lutwak's Logarithmic Minkowski conjecture about the uniqueness of even solution in the case are surveyed and connections to many related problems are discussed.
Paper Structure (5 sections, 5 theorems, 62 equations)

This paper contains 5 sections, 5 theorems, 62 equations.

Table of Contents

  1. Introduction
  2. Classical Brunn-Minkowski Theory
  3. Cone volume measure, log-Minkowski problem, log-Brunn-Minkowski conjecture
  4. Lutwak's $L_p$-Minkowski theory
  5. Some variants of the $L_p$-Minkowski problem

Key Result

THEOREM 2.1

For $\gamma^*(n)>0$ depending on $n$ and any convex bodies $K$ and $C$ in $\mathbb{R}^n$,

Theorems & Definitions (11)

  • THEOREM 2.1: Figalli, Maggi, Pratelli
  • LEMMA 2.2: Alexandrov
  • THEOREM 2.3: Prékopa-Leindler
  • CONJECTURE 3.1: Log-Minkowski conjecture #1
  • CONJECTURE 3.2: Log-Minkowski conjecture #2
  • THEOREM 3.3
  • CONJECTURE 3.4: Log-Brunn-Minkowski conjecture
  • CONJECTURE 4.1: $L_p$-Minkowski Conjecture #1
  • CONJECTURE 4.2: $L_p$-Minkowski conjecture #2
  • CONJECTURE 4.3: $L_p$-Brunn-Minkowski conjecture
  • ...and 1 more