Optimal stability of Pál's isominwidth inequality for ball convex bodies in planes of constant curvature
Abstract
Pál's isominwidth inequality (1921) answered the Kakeya needle problem (1917) for convex sets. It states that among convex bodies of fixed minimum width w in the Euclidean plane, the regular triangle has minimal area. The isominwidth inequality was generalized to the $2$-dimensional sphere by Bezdek and Blekherman and Freyer and Sagmeister (arXiv:2411.11462). Interestingly, in hyperbolic space, no minimizer exists, as shown by Böröczky, Freyer and Sagmeister (arXiv:2502.04427). The stability of the Euclidean Pál inequality with respect to the Hausdorff metric and the symmetric difference metric was proved by Lucardesi and Zucco (arXiv:2405.18294). Fodor, Robock and Sagmeister (arXiv:2602.19300) proved r-ball convex analogs of the isominwidth inequality in all three constant curvature planes connecting Pál's theorem with the Blaschke--Lebesgue inequality. In this paper, we prove optimal stability versions of this statement with respect to the Hausdorff distance and the symmetric difference metric in all three constant curvature planes.
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BibTeX
@misc{fodor2026optimal,
title = {{Optimal stability of Pál's isominwidth inequality for ball convex bodies in planes of constant curvature}},
author = {Ferenc Fodor and Ádám Sagmeister},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2606.02882},
eprint = {2606.02882},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2606.02882},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Optimal stability of Pál's isominwidth inequality for ball convex bodies in planes of constant curvature
AU - Ferenc Fodor
AU - Ádám Sagmeister
PY - 2026
DA - 2026/10/06
PB - arXiv
DO - 10.48550/arxiv.2606.02882
UR - https://arxiv.org/abs/2606.02882
AB - Pál's isominwidth inequality (1921) answered the Kakeya needle problem (1917) for convex sets. It states that among convex bodies of fixed minimum width w in the Euclidean plane, the regular triangle has minimal area. The isominwidth inequality was generalized to the $2$-dimensional sphere by Bezdek and Blekherman and Freyer and Sagmeister (arXiv:2411.11462). Interestingly, in hyperbolic space, no minimizer exists, as shown by Böröczky, Freyer and Sagmeister (arXiv:2502.04427). The stability of the Euclidean Pál inequality with respect to the Hausdorff metric and the symmetric difference metric was proved by Lucardesi and Zucco (arXiv:2405.18294). Fodor, Robock and Sagmeister (arXiv:2602.19300) proved r-ball convex analogs of the isominwidth inequality in all three constant curvature planes connecting Pál's theorem with the Blaschke--Lebesgue inequality. In this paper, we prove optimal stability versions of this statement with respect to the Hausdorff distance and the symmetric difference metric in all three constant curvature planes.
ER -