We investigate a convexity properties for normalized log moment generating function continuing a recent investigation of Chen of convex images of Gaussians. We show that any variable satisfying a ``Ehrhard-like'' property for its distribution function has a strictly convex normalized log moment generating function, unless the variable is Gaussian, in which case affine-ness is achieved. Moreover we characterize variables that satisfy the Ehrhard-like property as the convex images of Gaussians. As applications, we derive sharp comparisons between Rényi divergences for a Gaussian and a strongly log-concave variable, and characterize the equality case. We also demonstrate essentially optimal concentration bounds for the sequence of conic intrinsic volumes associated to convex cone and we obtain a reversal of McMullen's inequality between the sum of the (Euclidean) intrinsic volumes associated to a convex body and the body's mean width that generalizes and sharpens a result of Alonso-Hernandez-Yepes.