  We establish a new comparison between the Legendre transform of the cumulant
generating function and the half-space depth of an arbitrary log-concave
probability distribution on the real line, that carries on to the
multidimensional setting. Combined with sharp estimates for the Cram\'{e}r
transform of rotationally invariant measures, we are led to some new
phase-transition type results for the asymptotics of the expected measure of
random polytopes. As a byproduct of our analysis, we address a question on the
sharp exponential separability constant for log-concave distributions, in the
symmetric case.
