Year: 2026
Type: preprint
Abstract: We prove that every convex body \(K\subset\mathbb{R}^n\) admits \(S\in\mathrm{SL}_n(\mathbb{R})\) 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 maximal Gaussian measure position. The main step is the spectral estimate\[\lambda_1(D)\le C(P)\bigl(n+\sqrt{n I(\nu\mid\gamma_n)}\bigr)\]for an even isotropic log-concave probability measure \(\nu\) supported on a symmetric convex body \(D\), with Poincar\'e 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 \(\lambda_1(K)\). The square-root dependence then gives the bound of order \(n\).
Source: arXiv (Cornell University)
Authors: Colbrook, Matthew J., Stepaniants, George
Institution: University of Cambridge
Language: English
Subfield: Applied Mathematics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7225017002