  It is well known that a log-Sobolev inequality implies sub-gaussian decay of
the tails. In the spirit of the KLS conjecture, we investigate whether this
implication can be reversed under a log-concavity assumption. In the general
setting, we improve on a result of Bobkov, establishing the best dimension
dependent bound on the log-Sobolev constant of subgaussian log-concave
measures, and we investigate some special cases.
