Self-dual smoothing of convex and saddle functions

It is shown that any convex function can be approximated by a family of differentiable with Lipschitz continuous gradient and strongly convex approximates in a “self-dual” way: the conjugate of each approximate is the approximate of the conjugate of the original function. The approximation technique extends to saddle functions, and is self-dual with respect to saddle function conjugacy and also partial conjugacy that relates saddle functions to convex functions. Mathemtatics Subject Classification. 52A41, 90C25, 90C59, 90C46, 26B25