The Ashbaugh--Benguria reciprocal-gap conjecture for Dirichlet eigenvalues

We prove the Ashbaugh–Benguria reciprocal-gap conjecture for the Dirichlet Laplacian in every dimension $N\ge 2$. Specifically, if $\mathrm{\Omega }\subset {\mathbb{R}}^N$ is a bounded domain and $$0<{\lambda }_1(\mathrm{\Omega })<{\lambda }_2(\mathrm{\Omega })\le {\lambda }_3(\mathrm{\Omega })\le \cdots$$ are its Dirichlet eigenvalues, then $$\sum _{i=1}^N\frac{{\lambda }_1(\mathrm{\Omega })}{{\lambda }_{i+1}(\mathrm{\Omega })-{\lambda }_1(\mathrm{\Omega })}\ge \frac{N}{j_{N\mathrm{/}2,1}^2\mathrm{/}{j}_{N\mathrm{/}2-1,1}^2-1},$$ where $j_{\mu ,1}$ denotes the first positive zero of the Bessel function $J_{\mu }$ of the first kind of order $\mu$. We also characterize the equality case: equality holds precisely when $\mathrm{\Omega }$ agrees with a Euclidean ball up to a set of Sobolev $H^1$-capacity zero. In particular, among bounded Lipschitz domains, equality holds if and only if $\mathrm{\Omega }$ is a Euclidean ball.