We show that the Laplace approximation of a supremum by $L^p$-norms has interesting consequences in optimization. For instance, the logarithmic barrier functions (LBF) of a primal convex problem $P$ and its dual $P^*$ appear naturally when using this simple approximation technique for the value function $g$ of $P$ or its Legendre-Fenchel conjugate $g^*$. In addition, minimizing the LBF of the dual $P^*$ is just evaluating the Cramer transform of the Laplace approximation of $g$. Finally, this technique permits to sometimes define an explicit dual problem $P^*$ in cases when the Legendre-Fenchel conjugate $g^*$ cannot be derived explicitly from its definition.
View ^phBcnorms, log-barriers and Cramer transform in optimization