An affine reverse Faber-Krahn inequality via information decay

Abstract

We prove that every convex body $K\subset\mathbb{R}^n$ admits $S\in\mathrm{SL}_n(\mathbb{R})$ such that $λ_1(SK)|K|^{2/n}\le Cn$, where $C$ is universal. For origin-symmetric bodies of volume one, the bound holds in maximal Gaussian measure position. The main step is the spectral estimate \[ λ_1(D)\le C(P)\bigl(n+\sqrt{n I(ν\midγ_n)}\bigr) \] for an even isotropic log-concave probability measure $ν$ supported on a symmetric convex body $D$, with Poincaré constant at most $P$ and finite relative Fisher information. A second-order Stein estimate and an information jump under convolution give entropy decay faster than $m^{-1}$ for normalised sums. Their supports control the Gaussian measure of dilates, and a Brownian bridge expansion gives the spectral estimate. A Gaussian-weighted ground state supplies a measure whose Fisher information is bounded in terms of $λ_1(K)$. The square-root dependence then gives the bound of order $n$.