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

Functional Analysis