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

TL;DR

This paper establishes a specific threshold value p_0 in the range [0,1) for the even L^p Minkowski problem, proving uniqueness of solutions for p above p_0 and demonstrating non-uniqueness for p below p_0, thus refining the understanding of solution behavior.

Contribution

The paper identifies a critical eigenvalue-based parameter p_0 that determines the uniqueness of solutions in the even L^p Minkowski problem, extending previous results.

Findings

Uniqueness holds for p ≥ p_0

Non-uniqueness occurs for infinitely many convex bodies when p < p_0

p_0 is characterized by the eigenvalue of a Hilbert operator

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 $p>p_0$.