Sharp stability for the (B)-theorem

Abstract

The (B)-theorem of Cordero-Erausquin, Fradelizi and Maurey states that if $\gamma$ is the standard Gaussian in ${\mathbb{R}}^n$, $K\subset {\mathbb{R}}^n$ is an origin-symmetric convex set, and $s,t\in \mathbb{R}$ then $\gamma \left(e^{\frac{s+t}{2}}K\right)\ge \sqrt{\gamma (e^{s}K)\gamma (e^{t}K)}$. Herscovici, Livshyts, Rotem and Volberg proved a stability version of this result, showing that if one has equality up to a factor $(1+\epsilon )$ in the (B)-inequality for $K$ then the inradius of $K$ must be either ``very large'' or ``very small,'' where the bounds depend on $\epsilon$ and on $n$. We give a new stability estimate which is dimension-free and also yields more precise information about bodies which are near-optimizers of the (B)-inequality. In particular, our results imply that if $\gamma \left(e^{\frac{s+t}{2}}K\right)\le (1+\epsilon )\sqrt{\gamma (e^{s}K)\gamma (e^{t}K)}$, then every principal component of the covariance matrix of the probability measure obtained by restricting the Gaussian to $K$ must either be at least $1-O(\epsilon )$ or at most $O(\epsilon )$, which is sharp. Our method extends immediately to yield stability estimates for generalizations of the (B)-inequality, namely the ``strong'' and ``functional'' (B)-inequalities, which reduce to spectral questions about $1$-log-concave measures on ${\mathbb{R}}^n$.

16 pages