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In this work, we construct a proximal average for two prox-bounded functions,which recovers the classical proximal average for two convex functions. The newproximal average transforms continuously in epi-topology from one proximal hullto the other. When one of the functions is differentiable, the new proximalaverage is differentiable. We give characterizations for Lipschitz andsingle-valued proximal mappings and we show that the convex combination ofconvexified proximal mappings is always a proximal mapping.Subdifferentiability and behaviors of infimal values and minimizers are alsostudied.

A proximal average for prox-bounded functions