An affine reverse Faber-Krahn inequality via information decay

We prove that every convex body \(K\subset\R^n\) admits\(S\in\SL_n(\R)\) such that\(\lambda_1(SK)|K|^{2/n}\le Cn\), where \(C\) is universal.For origin-symmetric bodies of volume one, the bound holds in maximalGaussian measure position. The main step is the spectral estimate\[\lambda_1(D)\le C(P)\bigl(n+\sqrt{n I(\nu\mid\gam)}\bigr)\]for an even isotropic log-concave probability measure \(\nu\) supported ona symmetric convex body \(D\), with Poincar\'e constant at most \(P\) andfinite relative Fisher information.A second-order Stein estimate and an information jump under convolutiongive entropy decay faster than \(m^{-1}\) for normalised sums. Theirsupports control the Gaussian measure of dilates, and a Brownian bridgeexpansion gives the spectral estimate. A Gaussian-weighted ground statesupplies a measure whose Fisher information is bounded in terms of\(\lambda_1(K)\). The square-root dependence then gives the bound oforder \(n\).