The weighted $L^p$ Minkowski problem

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–Zhang, and weighted Minkowski problems, introduced by Livshyts. For the latter, the Gaussian Minkowski problem, whose primary investigators were Huang–Xi–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 obtain uniqueness for $p\ge 1$ 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–Zhao.