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 R\'enyi 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.

翻译：我们研究归一化对数矩生成函数的凸性性质，延续了Chen近期关于高斯凸像的研究。我们证明，任何满足分布函数"类Ehrhard"性质的变量都具有严格凸的归一化对数矩生成函数，除非该变量是高斯变量（此时达到仿射性）。此外，我们将满足类Ehrhard性质的变量表征为高斯凸像。作为应用，我们推导了高斯变量与强对数凹变量之间Rényi散度的尖锐比较，并刻画了等号成立的情形。我们还证明了与凸锥相关的圆锥内蕴体积序列本质上最优的集中界，并获得了凸体关联的（欧几里得）内蕴体积之和与体平均宽度之间McMullen不等式的反向形式，该结果推广并强化了Alonso-Hernandez-Yepes的结论。