Geodesic
A saddle-point theorem for strongly and weakly convex functions
Izvestiya. Mathematics , Tome 75 (2011) no. 1, pp. 73-100.

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We prove a theorem on the existence, uniqueness, and continuous dependence on parameters for a saddle point in a type of minimax problem that arises, for example, in differential game theory. Our theorem on the existence of a saddle point does not follow from the well-known theorems of von Neumann, Ky Fan, Sion and others since the intersection of sublevel sets of the function considered may be disconnected and non-empty. The hypotheses of our theorem are stated in terms of the strong and weak convexity of functions defined on a Banach space. We study properties of strongly and weakly convex functions related to the operations of minimization and maximization. We obtain unimprovable estimates of convexity parameters for the infimal convolution (episum) and epidifference of functions. This results in the construction of a calculus of convexity parameters of functions with respect to epioperations. We give typical examples and show that the hypotheses of our theorems are essential.
Keywords: saddle point, minimax, strong and weak convexity, differential game. 
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G. E. Ivanov. A saddle-point theorem for strongly and weakly convex functions. Izvestiya. Mathematics , Tome 75 (2011) no. 1, pp. 73-100. http://geodesic.mathdoc.fr/item/IM2_2011_75_1_a3/

[1] E. S. Polovinkin, M. V. Balashov, Elementy vypuklogo i silno vypuklogo analiza, 1-e izd., Fizmatlit, M., 2004; 2-е изд., испр и доп.; Физматлит, М., 2007 | Zbl

[2] G. E. Ivanov, Slabo vypuklye mnozhestva i funktsii, Fizmatlit, M., 2006 | Zbl

[3] B. T. Poljak, “Existence theorems and convergence of minimizing sequences in extremum problems with restrictions”, Soviet Math. Dokl., 7 (1966), 72–75 | MR | Zbl

[4] J. von Neumann, “On the theory of games of strategy”, Contributions to the theory of games, v. IV, Princeton Univ. Press, Princeton, NJ, 1959, 13–42 | MR | MR | Zbl

[5] K. Fan, “Fixed-point and minimax theorems in locally convex topological linear spaces”, Proc. Nat. Acad. Sci. USA, 38:2 (1952), 121–126 | DOI | MR | Zbl

[6] M. Sion, “On general minimax theorems”, Pacific J. Math., 8 (1958), 171–176 | MR | Zbl

[7] J.-J. Moreau, “Inf-convolution des fonctions numeriques sur un espace vectoriel”, C. R. Acad. Sci. Paris, 256 (1963), 5047–5049 | MR | Zbl

[8] E. S. Polovinkin, “On strongly convex sets and strongly convex functions”, J. Math. Sci. (New York), 100:6 (2000), 2633–2681 | DOI | MR | Zbl

[9] N. N. Krasovskii, Upravlenie dinamicheskoi sistemoi, Nauka, M., 1985 | MR | Zbl

[10] L. S. Pontrjagin, “Linear differential games of pursuit”, Math. USSR-Sb., 40:3 (1981), 285–303 | DOI | MR | Zbl | Zbl

[11] I. Ekeland, R. Temam, Convex analysis and variational problems, Studies in Mathematics and its Applications, 1, North-Holland, Amsterdam–Oxford; Elsevier, New York, 1976 | MR | MR | Zbl | Zbl