Abstract convex optimal antiderivatives
Abstract
Having studied families of antiderivatives and their envelopes in the setting of classical convex analysis, we now extend and apply these notions and results in settings of abstract convex analysis. Given partial data regarding a csubdifferential, we consider the set of all cconvex cantiderivatives that comply with the given data. Under a certain assumption, this set is not empty and contains both its lower and upper envelopes. We represent these optimal antiderivatives by explicit formulae. Some well known functions are, in fact, optimal cconvex cantiderivatives. In one application, we point out a natural minimality property of the Fitzpatrick function of a cmonotone mapping, namely that it is a minimal antiderivative. In another application, in metric spaces, a constrained Lipschitz extension problem fits naturally the convexity notions we discuss here. It turns out that the optimal Lipschitz extensions are precisely the optimal antiderivatives. This approach yields explicit formulae for these extensions, the most particular case of which recovers the well known extensions due to McShane and Whitney.
 Publication:

arXiv eprints
 Pub Date:
 January 2014
 arXiv:
 arXiv:1401.6869
 Bibcode:
 2014arXiv1401.6869B
 Keywords:

 Mathematics  Functional Analysis;
 Mathematics  Optimization and Control;
 47H04 47H05 49N15 52A01
 EPrint:
 Annales de l'Institut Henri Poincare (C) Analyse Non Lineaire 29 (2012) 435454