Abstract Convexity and
S. S. Kutateladze
The slides are available in
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.
Fenchel W. (1953)
Convex Cones, Sets, and Functions. Princeton: Princeton Univ. Press.
Hörmander L. (1955)
Sur la fonction d'appui des ensembles convexes dans une espace lokalement
convexe. Ark. Mat., 3:2, 180–186 [in French].
Hörmander L. (1994)
Notions of Convexity. Boston: Birkhäuser.
Kutateladze S. S. and Rubinov A. M. (1972)
Minkowski duality and its applications.
Russian Math. Surveys, 27:3, 137–191.
Kutateladze S. S. and Rubinov A. M. (1976)
Minkowski Duality and Its Applications.
Novosibirsk: Nauka Publishers [in Russian].
Singer I. (1997)
Abstract Convex Analysis.
New York: John Wiley & Sons.
Pallaschke D. and Rolewicz S. (1998)
Foundations of Mathematical Optimization, Convex
Analysis Without Linearity. Dordrecht: Kluwer Academic Publishers.
Rubinov A. M. (2000)
Abstract Convexity and Global Optimization. Dordrecht:
Kluwer Academic Publishers.
Ioffe A. D. and Rubinov A. M. (2002)
Abstract convexity and nonsmooth analysis. Global aspects.
Adv. Math. Econom., 4, 1–23.
Fuchssteiner B. and Lusky W. (1981)
Convex Cones. Amsterdam: North-Holland.
Kusraev A. G. and Kutateladze S. S. (2007)
Subdifferential Calculus: Theory and Applications.
Moscow: Nauka Publishers [in Russian].
Dilworth S. J., Howard R., and Roberts J. W. (2006)
A general theory of almost convex functions. Trans. Amer. Math. Soc.,
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.”
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On 18 May 2007, 09:01.