The Weighted $L^p$ Minkowski Problem

TL;DR

This paper extends the $L^p$ Minkowski problem to weighted surface measures with rotational invariance, establishing existence, uniqueness, and small mass results across various $p$ values.

Contribution

It introduces a new framework for weighted $L^p$ Minkowski problems beyond Gaussian measures, including existence and uniqueness results under symmetry and concavity assumptions.

Findings

Existence of solutions for all real p with symmetry assumptions in some cases.

Uniqueness of solutions for p ≥ 1 under concavity conditions.

Results on small mass regimes using degree theory.

Abstract

The Minkowski problem in convex geometry concerns showing that a given Borel measure on the unit sphere is, up to perhaps a constant, some type of surface area measure of a convex body. Two types of Minkowski problems in particular are an active area of research: $L^p$ Minkowski problems, introduced by Lutwak and (Lutwak, Yang, and Zhang), and weighted Minkowski problems, introduced by Livshyts. For the latter, the Gaussian Minkowski problem, whose primary investigators were (Huang, Xi and Zhao), is the most prevalent. In this work, we consider weighted surface area in the $L^p$ setting. We propose a framework going beyond the Gaussian setting by focusing on rotationally invariant measures, mirroring the recent development of the Gardner-Zvavitch inequality for rotationally invariant, log-concave measures. Our results include existence for all $p \in \mathbb R$ (with symmetry assumptions in certain instances). We also have uniqueness for $p \geq 1$ under a concavity assumption. Finally, we obtain results in the so-called "small mass regime" using degree theory, as instigated in the Gaussian case by (Huang, Xi and Zhao).