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arXiv · 2610.28757

Sharp quantitative stability of the Hong--Krahn--Szegő inequality

Yuxuan Zhou·Wenming Zou
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Abstract

We establish sharp quantitative stability estimates for the Hong--Krahn--Szegő inequality for arbitrary open sets of positive finite measure in $\mathbb R^d$, $d\ge2$. In dimensions $d\ge3$, 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}.

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Yuxuan Zhou, Wenming Zou. 2026-10-20. Sharp quantitative stability of the Hong--Krahn--Szegő inequality. https://arxiv.org/abs/2610.28757

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