Abstract convex optimal antiderivatives
Annales de l'I.H.P. Analyse non linéaire, Volume 29 (2012) no. 3, p. 435-454

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.

DOI : https://doi.org/10.1016/j.anihpc.2012.01.004
Classification:  47H04,  47H05,  49N15,  52A01
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},
     publisher = {Elsevier},
     volume = {29},
     number = {3},
     year = {2012},
     pages = {435-454},
     doi = {10.1016/j.anihpc.2012.01.004},
     zbl = {1259.47064},
     mrnumber = {2926243},
     language = {en},
     url = {http://www.numdam.org/item/AIHPC_2012__29_3_435_0}
}
Bartz, Sedi; Reich, Simeon. Abstract convex optimal antiderivatives. Annales de l'I.H.P. Analyse non linéaire, Volume 29 (2012) no. 3, pp. 435-454. doi : 10.1016/j.anihpc.2012.01.004. http://www.numdam.org/item/AIHPC_2012__29_3_435_0/

[1] G. Aronsson, M.G. Crandall, P.A. Juutinen, A tour of the theory of absolutely minimizing functions, Bull. Amer. Math. Soc. 41 (2004), 439-505 | MR 2083637 | Zbl 1150.35047

[2] S. Bartz, S. Reich, Minimal antiderivatives and monotonicity, Nonlinear Anal. 74 (2011), 59-66 | MR 2734977 | Zbl 1251.47046

[3] H. Brezis, Liquid crystals and energy estimates for S 2 -valued maps, Theory and Applications of Liquid Crystals, Minneapolis, Minn., 1985, IMA Vol. Math. Appl. vol. 5, Springer, New York (1987), 31-52 | MR 900828

[4] H. Brezis, J.-M. Coron, E.H. Lieb, Harmonic maps with defects, Comm. Math. Phys. 107 (1986), 649-705 | MR 868739 | Zbl 0608.58016

[5] R.S. Burachik, A. Rubinov, Abstract convexity and augmented Lagrangians, SIAM J. Optim. 18 (2007), 413-436 | MR 2338445 | Zbl 1190.90134

[6] J.J.M. Evers, H. Van Maaren, Duality principles in mathematics and their relations to conjugate functions, Nieuw Arch. Wiskunde 3 (1985), 23-68 | MR 786636 | Zbl 0598.49009

[7] S. Fitzpatrick, Representing monotone operators by convex functions, Workshop/Miniconference on Functional Analysis and Optimization, Canberra, 1988, Proceedings of the Centre for Mathematical Analysis, vol. 20, Australian National University, Canberra, Australia (1988), 59-65 | MR 1009594

[8] J.-B. Hiriart-Urruty, Extension of Lipschitz functions, J. Math. Anal. Appl. 77 (1980), 539-554 | MR 593233 | Zbl 0455.26006

[9] J.E. Martínez-Legaz, On lower subdifferentiable functions, Trends in Mathematical Optimization, Birkhäuser, Basel (1988), 197-232 | MR 1017954 | Zbl 0643.49015

[10] E.J. Mcshane, Extension of range of functions, Bull. Amer. Math. Soc. 40 (1934), 837-842 | MR 1562984 | Zbl 0010.34606

[11] J.J. Moreau, Inf-convolution, sous-additivité, convexité des fonctions numériques, J. Math. Pures Appl. 49 (1970), 109-154 | MR 288602 | Zbl 0195.49502

[12] J.-P. Penot, Monotonicity and dualities, Generalized Convexity and Related Topics, Springer, Berlin (2006), 399-414 | MR 2279416

[13] R.T. Rockafellar, Characterization of the subdifferentials of convex functions, Pacific J. Math. 17 (1966), 497-510 | MR 193549 | Zbl 0145.15901

[14] R.T. Rockafellar, On the maximal monotonicity of subdifferential mappings, Pacific J. Math. 33 (1970), 209-216 | MR 262827 | Zbl 0199.47101

[15] S. Rolewicz, Φ-convex functions defined on metric spaces, J. Math. Sci. 115 (2003), 2631-2652 | MR 1992986 | Zbl 1057.49017

[16] A.M. Rubinov, Abstract Convexity and Global Optimization, Kluwer, Dordrecht (2000) | MR 1834382 | Zbl 0963.49014

[17] I. Singer, Abstract Convex Analysis, Wiley–Interscience, New York (1997) | MR 1461544 | Zbl 0898.49001

[18] C. Villani, Optimal Transport: Old and New, Springer, Berlin (2009) | MR 2459454 | Zbl 1156.53003

[19] H. Whitney, Analytic extensions of differentiable functions defined in closed sets, Trans. Amer. Math. Soc. 36 (1934), 63-89 | JFM 60.0217.01 | MR 1501735 | Zbl 0008.24902