On uniformly convex functions

  1. Grelier, Guillaume
  2. Raja Baño, Matías 1
  1. 1 Universidad de Murcia
    info

    Universidad de Murcia

    Murcia, España

    ROR https://ror.org/03p3aeb86

Journal:
Journal of Mathematical Analysis and Applications

ISSN: 0022-247X

Year of publication: 2022

Volume: 505

Issue: 1

Pages: 125442

Type: Article

DOI: 10.1016/J.JMAA.2021.125442 GOOGLE SCHOLAR lock_openOpen access editor

More publications in: Journal of Mathematical Analysis and Applications

Abstract

Non-convex functions that yet satisfy a condition of uniform convexity for non-closepoints can arise in discrete constructions. We prove that this sort of discrete uniformconvexity is inherited by the convex envelope, which is the key to obtain otherremarkable properties such as the coercivity. Our techniques allow to retrieve Enflo’suniformly convex renorming of super-reflexive Banach spaces as the regularizationof a raw function built from trees. Among other applications, we provide a sharpestimation of the distance of a given function to the set of differences of Lipschitzconvex functions. Finally, we prove the equivalence of several possible ways toquantify the super weakly noncompactness of a convex subset of a Banach space.

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