The Entropic Barrier: Exponential Families, Log-Concave Geometry, and Self-Concordance

We prove that the Cramér transform of the uniform measure on a convex body in ℝⁿ 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. As a side result, our calculations also show that the universal barrier of Nesterov and Nemirovski is exactly n-self-concordant on convex cones.