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 \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–Zhao.

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BibTeX RIS

BibTeX

@article{langharst2026weighted,
  title = {{The weighted $L^p$ Minkowski problem}},
  author = {Dylan Langharst and Jiaqian Liu and Shengyu Tang},
  journal = {Revista Matemática Iberoamericana},
  year = {2026},
  month = jun,
  doi = {10.4171/rmi/1636},
  url = {https://doi.org/10.4171/rmi/1636},
  note = {CC BY},
}

RIS

TY  - JOUR
TI  - The weighted L^p Minkowski problem
AU  - Dylan Langharst
AU  - Jiaqian Liu
AU  - Shengyu Tang
PY  - 2026
DA  - 2026/06/19
JO  - Revista Matemática Iberoamericana
DO  - 10.4171/rmi/1636
UR  - https://doi.org/10.4171/rmi/1636
AB  - 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 \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–Zhao.
SN  - 0213-2230
SN  - 2235-0616
ER  -

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