The entropic barrier: a simple and optimal universal self-concordant   barrier
  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. 
S\'ebastien Bubeck and Ronen Eldan
math.OC cs.IT cs.LG math.IT