Abstract convex optimal antiderivatives

Bartz, Sedi; Reich, Simeon

Annales de l'I.H.P. Analyse non linéaire, Tome 29 (2012) no. 3, pp. 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.

[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 | Zbl

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

[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

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

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

[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 | Zbl

[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

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

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

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

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

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

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

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

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

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

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

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

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