On uniformly convex functions

Non-convex functions that yet satisfy a condition of uniform convexity for non-close points can arise in discrete constructions. We prove this sort is inherited by the convex envelope, which key to obtain other remarkable properties such as coercivity. Our techniques allow retrieve Enflo's uniformly renorming super-reflexive Banach spaces regularization raw function built from trees. Among applications, we provide sharp estimation distance given set differences Lipschitz functions. Finally, equivalence several possible ways quantify super weakly noncompactness subset space.