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 ${\mathbb{R}}^d$, $d\ge 2$. In dimensions $d\ge 3$, 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}.