Sharp quantitative stability of the Ashbaugh--Benguria inequality

We establish sharp quantitative stability of the Ashbaugh–Benguria inequality on bounded convex domains in ${\mathbb{R}}^n$. After volume and centroid normalization, the eigenvalue-ratio deficit controls the squared $H^{-1\mathrm{/}2}({\mathbb{S}}^{n-1})$ norm of the radial displacement from the unit sphere. This yields the optimal exponent $5\mathrm{/}2$ for the Fraenkel asymmetry and, for small deficits, the optimal exponent $(n+4)\mathrm{/}2$ for the normalized Hausdorff distance. All three exponents are sharp even among smooth strictly convex domains whose principal curvatures lie in $[1\mathrm{/}2,2]$. In particular, the quadratic Fraenkel-asymmetry estimate proposed by Brasco and De Philippis (2017, Open Problem 7.65) fails. We also determine the optimal stability exponents under higher-order radial regularity bounds.