A Proof of Bobkov's Spectral Bound For Convex Domains via Gaussian Fitting and Free Energy Estimation

TL;DR

This paper presents a novel proof of Bobkov's spectral bound for convex domains by using Gaussian measure comparison and free energy estimates, avoiding traditional localization techniques.

Contribution

It introduces a new proof method based on spectral transference and free energy bounds, differing from existing geometric localization approaches.

Findings

Established a lower bound on the first eigenvalue of the Neumann Laplacian for convex domains.

Demonstrated the effectiveness of Gaussian measure comparison in spectral bounds.

Provided sharper estimates on free energy of conditioned Gaussian measures.

Abstract

We obtain a new proof of Bobkov's lower bound on the first positive eigenvalue of the (negative) Neumann Laplacian (or equivalently, the Cheeger constant) on a bounded convex domain $K$ in Euclidean space. Our proof avoids employing the localization method or any of its geometric extensions. Instead, we deduce the lower bound by invoking a spectral transference principle for log-concave measures, comparing the uniform measure on $K$ with an appropriately scaled Gaussian measure which is conditioned on $K$. The crux of the argument is to establish a good overlap between these two measures (in say the relative-entropy or total-variation distances), which boils down to obtaining sharp lower bounds on the free energy of the conditioned Gaussian measure.