  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.
