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On the uniqueness of even $L^p$ Minkowski problem

Weiyong He, Junbang Liu

TL;DR

This work addresses the uniqueness of the even $L_p$ Minkowski problem by linking it to the spectrum of the Hibert–Brunn–Minkowski operator via the first even eigenvalue $\lambda_{1,e}(-L_K)$. It introduces a sharp threshold $p_0\in[0,1)$ characterized by $\inf_{K\in\mathcal{K}^2_{+,e}} \lambda_{1,e}(-L_K)=n+1-p_0$, and proves uniqueness for $p\ge p_0$ (including $p_0$ itself) while demonstrating non-uniqueness for $p<p_0$ in general. The results hinge on a variational formulation for the Monge–Ampère type equation on $S^n$, the local $L_p$-Minkowski conjecture, and a strict eigenvalue bound $\lambda_{1,e}(-L_K)>n+1-p_0$ for smooth, origin-symmetric bodies, with extensions to nonsmooth convex bodies via a complex-analytic framework. Collectively, the findings connect even $L_p$ Brunn–Minkowski inequalities, local conjectures, and the global uniqueness landscape for the even $L_p$ Minkowski problem, advancing the theory toward a complete classification at the critical threshold $p_0$.

Abstract

We prove that there is a unique $p_0\in [0,1)$, which can be characterized by the eigenvalue of Hilbert operator related to a convex body, that the even $L^p$ Minkowski problem has a unique solution for $p\geq p_0$, and the uniqueness fails for infinitely many convex bodies if $p<p_0$. The previous results by many experts in the field assert that the uniqueness holds for $p>p_0$.

On the uniqueness of even $L^p$ Minkowski problem

TL;DR

This work addresses the uniqueness of the even Minkowski problem by linking it to the spectrum of the Hibert–Brunn–Minkowski operator via the first even eigenvalue . It introduces a sharp threshold characterized by , and proves uniqueness for (including itself) while demonstrating non-uniqueness for in general. The results hinge on a variational formulation for the Monge–Ampère type equation on , the local -Minkowski conjecture, and a strict eigenvalue bound for smooth, origin-symmetric bodies, with extensions to nonsmooth convex bodies via a complex-analytic framework. Collectively, the findings connect even Brunn–Minkowski inequalities, local conjectures, and the global uniqueness landscape for the even Minkowski problem, advancing the theory toward a complete classification at the critical threshold .

Abstract

We prove that there is a unique , which can be characterized by the eigenvalue of Hilbert operator related to a convex body, that the even Minkowski problem has a unique solution for , and the uniqueness fails for infinitely many convex bodies if . The previous results by many experts in the field assert that the uniqueness holds for .
Paper Structure (4 sections, 16 theorems, 157 equations)

This paper contains 4 sections, 16 theorems, 157 equations.

Table of Contents

  1. Introduction
  2. The Minkowski problem and variational approach
  3. The proof of the main theorem
  4. The first even eigenvalue

Key Result

Theorem 1

There exists $p_n\in (0, 1)$ of the form $p_n=1-cn^{-3/2}$ such that, for all $K\in {\mathcal{K}}_{+, e}^2$, Moreover suppose hessianestimate holds for some fixed $n, p\in[0,1)$, and $K\in {\mathcal{K}}_{+,e}^2$, then for any $q>p$, there exists a $C^2$-neighborhood $N_{q,K}$ of $K$ in ${\mathcal{K}}_{+,e}^2$, such that if $L_1,L_2\in N_{q,K}$ satisfy $h_{L_1}^{1-q}dS_{L_1}=h_{L_2}^{1-q}dS_{L_2}$

Theorems & Definitions (28)

  • Conjecture 1: Böröczky-Lutwak-Yang-Zhang, Even $L_p$ Brunn-Minkowski conjecture
  • Conjecture 2
  • Conjecture 3: Local $L_p$-Minkowski conjecture, Kolesnikov-Milman
  • Conjecture 4: Local log-Minkowski conjecture, Kolesnikov-Milman
  • Theorem 1: Kolesnikov-Milman
  • Theorem 2: Chen-Huang-Li-Liu
  • Theorem 3
  • Theorem 4
  • Theorem 5
  • Conjecture 5
  • ...and 18 more