  We prove that the Cram\'er transform of the uniform measure on a convex body
in $\mathbb{R}^n$ is a $(1+o(1)) n$-self-concordant barrier, improving a
seminal result of Nesterov and Nemirovski. This gives the first explicit
construction of a universal barrier for convex bodies with optimal
self-concordance parameter. The proof is based on basic geometry of log-concave
distributions, and elementary duality in exponential families.
