On the geometry of convex typical sets

We consider convex sets obtained as one-sided typical sets of log-concave distributions, and show that the sequence of logarithms of intrinsic volumes corresponding to these typical sets converges to a limit function under an appropriate scaling. The limit function may be used to represent the exponential growth rate of intrinsic volumes of the typical sets. Since differential entropy is the exponential growth rate of the volume of typical sets, the exponential growth rate of intrinsic volumes generalizes the differential entropy of log-concave distributions. We conjecture a version of the entropy power inequality for such a generalization of differential entropy.