Inf-convolution and regularization of convex functions on Riemannian   manifolds of nonpositive curvature
  We show how an operation of inf-convolution can be used to approximate convex functions with $C^{1}$ smooth convex functions on Riemannian manifolds with nonpositive curvature (in a manner that not only is explicit but also preserves some other properties of the original functions, such as ordering, symmetries, infima and sets of minimizers), and we give some applications. 
Daniel Azagra and Juan Ferrera