  We prove upper bounds on the graph diameters of polytopes in two settings.
The first is a worst-case bound for polytopes defined by integer constraints in
terms of the height of the integers and certain subdeterminants of the
constraint matrix, which in some cases improves previously known results. The
second is a smoothed analysis bound: given an appropriately normalized
polytope, we add small Gaussian noise to each constraint. We consider a natural
geometric measure on the vertices of the perturbed polytope (corresponding to
the mean curvature measure of its polar) and show that with high probability
there exists a "giant component" of vertices, with measure $1-o(1)$ and
polynomial diameter. Both bounds rely on spectral gaps -- of a certain
Schr\"odinger operator in the first case, and a certain continuous time Markov
chain in the second -- which arise from the log-concavity of the volume of a
simple polytope in terms of its slack variables.
