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PreprintCC BY 4.0arXiv

On Santaló's Problem

  • Fang Hong
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Abstract

This paper investigates an isoperimetric-type problem posed by L. A. Santaló, concerning convex surfaces in hyperbolic 3-space that minimize total mean curvature among all convex surfaces with fixed surface area. This problem asks for a characterization of the minimizers and for the optimal form of a Minkowski-type inequality in hyperbolic 3-space. In this work, we propose a conjectural description of the minimizers under certain regularity assumptions. We also construct a new family of convex surfaces as as potential minimizer candidates and establish a property of the singular points of any minimizer.

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Cite

BibTeXRIS
BibTeX
@misc{hong2026santalós,
  title = {{On Santaló's Problem}},
  author = {Fang Hong},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2606.01771},
  eprint = {2606.01771},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2606.01771},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - On Santaló's Problem
AU  - Fang Hong
PY  - 2026
DA  - 2026/10/09
PB  - arXiv
DO  - 10.48550/arxiv.2606.01771
UR  - https://arxiv.org/abs/2606.01771
AB  - This paper investigates an isoperimetric-type problem posed by L. A. Santaló, concerning convex surfaces in hyperbolic 3-space that minimize total mean curvature among all convex surfaces with fixed surface area. This problem asks for a characterization of the minimizers and for the optimal form of a Minkowski-type inequality in hyperbolic 3-space. In this work, we propose a conjectural description of the minimizers under certain regularity assumptions. We also construct a new family of convex surfaces as as potential minimizer candidates and establish a property of the singular points of any minimizer.
ER  -