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Sharp quantitative stability of the Hong--Krahn--Szegő inequality
Abstract
We establish sharp quantitative stability estimates for the Hong--Krahn--Szegő inequality for arbitrary open sets of positive finite measure in ℝd, d≥2. In dimensions d≥3, the optimal stability exponent is (d+2)/2, while in dimension two the sharp stability modulus is t2/log(e/t). These results improve the quantitative stability estimate of Brasco and Pratelli~\cite[Theorem 3.5]{BP} and resolve Open Problem 7.56 of Brasco and De Philippis~\cite[Open Problem 7.56]{BDP}.
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BibTeX
@misc{zhou2026sharp,
title = {{Sharp quantitative stability of the Hong--Krahn--Szegő inequality}},
author = {Yuxuan Zhou and Wenming Zou},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.28757},
eprint = {2610.28757},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.28757},
}
RIS
TY - UNPB
TI - Sharp quantitative stability of the Hong--Krahn--Szegő inequality
AU - Yuxuan Zhou
AU - Wenming Zou
PY - 2026
DA - 2026/10/20
PB - arXiv
DO - 10.48550/arxiv.2610.28757
UR - https://arxiv.org/abs/2610.28757
AB - We establish sharp quantitative stability estimates for the Hong--Krahn--Szegő inequality for arbitrary open sets of positive finite measure in ℝ^d, d≥2. In dimensions d≥3, the optimal stability exponent is (d+2)/2, while in dimension two the sharp stability modulus is t^2/log(e/t). These results improve the quantitative stability estimate of Brasco and Pratelli~\cite[Theorem 3.5]{BP} and resolve Open Problem 7.56 of Brasco and De Philippis~\cite[Open Problem 7.56]{BDP}.
ER -