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

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.