Abstract Convexity and
Cone-Vexing Abstractions

S. S. Kutateladze

The slides are available in PDF.

This talk 1 is devoted to some origins of abstract convexity and a few vexing limitations on the range of abstraction in convexity. Convexity is a relatively recent subject. Although the noble objects of Euclidean geometry are mostly convex, the abstract notion of a convex set appears only after the Cantor paradise was founded. The idea of convexity feeds generation, separation, calculus, and approximation. Generation appears as duality; separation, as optimality; calculus, as representation; and approximation, as stability. Convexity is traceable from the remote ages and flourishes in functional analysis.

References

[1] Fenchel W. (1953) Convex Cones, Sets, and Functions. Princeton: Princeton Univ. Press.
[2] Hörmander L. (1955) Sur la fonction d'appui des ensembles convexes dans une espace lokalement convexe. Ark. Mat., 3:2, 180–186 [in French].
[3] Hörmander L. (1994) Notions of Convexity. Boston: Birkhäuser.
[4] Kutateladze S. S. and Rubinov A. M. (1972) Minkowski duality and its applications. Russian Math. Surveys, 27:3, 137–191.
[5] Kutateladze S. S. and Rubinov A. M. (1976) Minkowski Duality and Its Applications. Novosibirsk: Nauka Publishers [in Russian].
[6] Singer I. (1997) Abstract Convex Analysis. New York: John Wiley & Sons.
[7] Pallaschke D. and Rolewicz S. (1998) Foundations of Mathematical Optimization, Convex Analysis Without Linearity. Dordrecht: Kluwer Academic Publishers.
[8] Rubinov A. M. (2000) Abstract Convexity and Global Optimization. Dordrecht: Kluwer Academic Publishers.
[9] Ioffe A. D. and Rubinov A. M. (2002) Abstract convexity and nonsmooth analysis. Global aspects. Adv. Math. Econom., 4, 1–23.
[10] Fuchssteiner B. and Lusky W. (1981) Convex Cones. Amsterdam: North-Holland.
[11] Kusraev A. G. and Kutateladze S. S. (2007) Subdifferential Calculus: Theory and Applications. Moscow: Nauka Publishers [in Russian].
[12] Dilworth S. J., Howard R., and Roberts J. W. (2006) A general theory of almost convex functions. Trans. Amer. Math. Soc., 358:8, 3413–3445.

Footnote:

1 Delivered on August 26, 2007, at the International Workshop on the Idempotent and Tropical Mathematics in Moscow, August 26–30, 2007. I am grateful to Professor G. Litvinov who kindly invited this talk to the “tropics.”

Mathematics, abstract arXiv:0705.2793

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On 18 May 2007, 09:01.
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