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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 c-subdifferential, we consider the set of all c-convex c-antiderivatives 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 c-convex c-antiderivatives. In one application, we point out a natural minimality property of the Fitzpatrick function of a c-monotone 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.
Keywords: Abstract convexity, Convex function, Cyclically monotone operator, Fitzpatrick function, Lipschitz extension, Maximal monotone operator, Minimal antiderivative, Subdifferential operator
@article{AIHPC_2012__29_3_435_0,
author = {Bartz, Sedi and Reich, Simeon},
title = {Abstract convex optimal antiderivatives},
journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
pages = {435--454},
publisher = {Elsevier},
volume = {29},
number = {3},
year = {2012},
doi = {10.1016/j.anihpc.2012.01.004},
mrnumber = {2926243},
zbl = {1259.47064},
language = {en},
url = {http://geodesic.mathdoc.fr/articles/10.1016/j.anihpc.2012.01.004/}
}
TY - JOUR AU - Bartz, Sedi AU - Reich, Simeon TI - Abstract convex optimal antiderivatives JO - Annales de l'I.H.P. Analyse non linéaire PY - 2012 SP - 435 EP - 454 VL - 29 IS - 3 PB - Elsevier UR - http://geodesic.mathdoc.fr/articles/10.1016/j.anihpc.2012.01.004/ DO - 10.1016/j.anihpc.2012.01.004 LA - en ID - AIHPC_2012__29_3_435_0 ER -
%0 Journal Article %A Bartz, Sedi %A Reich, Simeon %T Abstract convex optimal antiderivatives %J Annales de l'I.H.P. Analyse non linéaire %D 2012 %P 435-454 %V 29 %N 3 %I Elsevier %U http://geodesic.mathdoc.fr/articles/10.1016/j.anihpc.2012.01.004/ %R 10.1016/j.anihpc.2012.01.004 %G en %F AIHPC_2012__29_3_435_0
Bartz, Sedi; Reich, Simeon. Abstract convex optimal antiderivatives. Annales de l'I.H.P. Analyse non linéaire, Tome 29 (2012) no. 3, pp. 435-454. doi: 10.1016/j.anihpc.2012.01.004
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