Spectral gap, logarithmic Sobolev constant, and geometric bounds

We survey recent works on the connection between spectral gap and logarithmic Sobolev constants, and exponential integrability of Lipschitz functions.In particular, tools from measure concentration are used to describe bounds on the diameter of a (compact) Riemannian manifold and of Markov chains in terms of the first eigenvalue of the Laplacian and the logarithmic Sobolev constant.We examine similarly dimension free isoperimetric bounds using these parameters.