The Weighted $L^p$ Minkowski Problem

Dylan Langharst, Jiaqian Liu, Shengyu Tang
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Abstract

The Minkowski problem in convex geometry concerns showing 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 rotational invariant measures, mirroring the recent development of the Gardner-Zvavitch inequality for rotational 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). Most known results for the Gaussian Minkowski problem are then special cases of our main theorems.
加权$L^p$闵科夫斯基问题
凸几何学中的闵科夫斯基问题涉及证明单位球面上的给定波罗测度是凸体的某种曲面面积测度,也许是一个常数。有两类闵科夫斯基问题尤其是一个活跃的研究领域:卢特瓦克和(卢特瓦克、杨和张)提出的 $L^p$ 闵科夫斯基问题,以及利夫希茨提出的加权闵科夫斯基问题。就后者而言,高斯闵科夫斯基问题最为普遍,其主要研究者是(黄、奚和赵)。在这项工作中,我们考虑的是 $L^p$ 背景下的加权表面积。我们提出了一个超越高斯背景的框架,重点关注旋转不变度量,反映了最近针对旋转不变对数凹度量的 Gardner-Zvavitch 不等式的发展。我们的结果包括所有 $p \in \mathbb R$ 的存在性(在某些情况下有对称性假设)。在凹性假设下,我们还得到了 $p \geq 1$ 的唯一性。最后,我们利用度数理论得到了所谓的$small$$mass$$regime$中的结果,正如(黄、奚、赵)在高斯情况下所启发的那样。高斯闵科夫斯基问题的大多数已知结果都是我们主要定理的特例。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
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