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PreprintCC BY 4.0arXiv

On the minimizer of the barycentric isoperimetric ratio among connected planar sets

  • Chiara Gambicchia
  • Aldo Pratelli
  • Clara Torres-Latorre
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Abstract

Among connected, planar sets, the ratio between the isoperimetric deficit and the squared barycentric asymmetry is known to be bounded from below by a strictly positive constant (as proved in recent papers by Bianchini-Croce-Henrot and Gambicchia-Pratelli). This opens the natural question whether the infimum is attained at some set. In this paper, we prove that a minimizer for this ratio actually exists, and moreover it is simply connected.

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Cite

BibTeXRIS
BibTeX
@misc{gambicchia2026minimizer,
  title = {{On the minimizer of the barycentric isoperimetric ratio among connected planar sets}},
  author = {Chiara Gambicchia and Aldo Pratelli and Clara Torres-Latorre},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.15088},
  eprint = {2610.15088},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.15088},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - On the minimizer of the barycentric isoperimetric ratio among connected planar sets
AU  - Chiara Gambicchia
AU  - Aldo Pratelli
AU  - Clara Torres-Latorre
PY  - 2026
DA  - 2026/10/10
PB  - arXiv
DO  - 10.48550/arxiv.2610.15088
UR  - https://arxiv.org/abs/2610.15088
AB  - Among connected, planar sets, the ratio between the isoperimetric deficit and the squared barycentric asymmetry is known to be bounded from below by a strictly positive constant (as proved in recent papers by Bianchini-Croce-Henrot and Gambicchia-Pratelli). This opens the natural question whether the infimum is attained at some set. In this paper, we prove that a minimizer for this ratio actually exists, and moreover it is simply connected.
ER  -