Moments of the Cramér transform of log-concave probability measures

Let $μ$ be a centered log-concave probability measure on ${\mathbb R}^n$ and let $Λ_μ^{\ast}$ denote the Cramér transform of $μ$, i.e. $Λ_μ^{\ast}(x)=\sup\{\langle x,ξ\rangle-Λ_μ(ξ):ξ\in\mathbb{R}^n\}$ where $Λ_μ$ is the logarithmic Laplace transform of $μ$. We show that $\mathbb{E}_μ\left[\exp\left(\frac{c_1}{n}Λ_μ^{\ast }\right)\right]<\infty $ where $c_1>0$ is an absolute constant. In, particular, $Λ_μ^{\ast}$ has finite moments of all orders. The proof, which is based on the comparison of certain families of convex bodies associated with $μ$, implies that $\|Λ_μ^{\ast}\|_{L^2(μ)}\leqslant c_2n\ln n$. The example of the uniform measure on the Euclidean ball shows that this estimate is optimal with respect to $n$ as the dimension $n$ grows to infinity.