  We introduce and investigate a new generalized convexity notion for functions
called prox-convexity. The proximity operator of such a function is
single-valued and firmly nonexpansive. We provide examples of (strongly)
quasiconvex, weakly convex, and DC (difference of convex) functions that are
prox-convex, however none of these classes fully contains the one of
prox-convex functions or is included into it. We show that the classical
proximal point algorithm remains convergent when the convexity of the proper
lower semicontinuous function to be minimized is relaxed to prox-convexity.
