Vexing Convexity

S. S. Kutateladze

1  Prolog

Convexity stems from the remote ages and reigns in geometry, optimization, and functional analysis. The union of abstraction and convexity has produced abstract convexity which is a vast area of today’s research, sometimes profitable but sometimes bizarre. The idea of convexity feeds generation, separation, calculus, and approximation. Generation appears as duality; separation, as optimality; calculus, as representation; and approximation, as stability. This talk is an overview of the origin, evolution, and basics of convexity, primarily emphasizing the rules of subdifferential calculus.

2  Origins of Convexity

2.1  Euclid’s Elements and Convexity

2.2  Harpedonaptae and Sulva Sutras

2.3  Enter Abstract Convexity

2.4  Mathesis and Abstraction

3  Convexity and Optimization

3.1  Minkowski Duality

3.2  Enter the Reals

3.3  Legendre in Disguise

3.4  Order Omnipresent

3.5  Fermat’s Criterion

4  Subdifferential Calculus

4.1  Enter Hahn–Banach

4.2  Enter Kantorovich

4.3  Ioffe’s Fans

4.4  Canonical Operator

4.5  Support Hull

4.6  Hahn–Banach in Disguise

5  Boolean Methods in Convexity

5.1  Enter Boole

5.2  Enter Descent

5.3  The Reals in Disguise

5.4  Approximation

5.5  Enter Epsilon and Monad

5.6  Subdifferential Halo

5.7  Exeunt Epsilon

6  Epilog

Abstract convexity starts with repudiating the heritage of harpedonaptae, which is annoying and vexing but may turn out rewarding. Freedom of set theory empowered us with the Boolean-valued models yielding various realizations a lot of surprising and unforeseen visualizations of the continuum. Many promising opportunities are open to modeling the powerful habits of reasoning and verification. Convexity is a topical illustration of the wisdom and strength of mathematics, the ever fresh art and science of calculus.

References

[1] Heaves Th. (1956) The Thirteen Books of Euclid’s Elements. Vol. 1–3. New York: Dover Publications.
[2] Kak S. C. (1997) Science in Ancient India. In: Sridhar S. R. and Mattoo N. K. (eds.) Ananya: A Portrait of India. New York: AIA, 399–420.
[3] Kutateladze S. S. and Rubinov A. M. (1972) Minkowski duality and its applications. Russian Math. Surveys, 27:3, 137–191.
[4] Kutateladze S. S. and Rubinov A. M. (1976) Minkowski Duality and Its Applications. Novosibirsk: Nauka Publishers [in Russian].
[5] Fuchssteiner B. and Lusky W. (1981) Convex Cones. Amsterdam: North-Holland.
[6] Hörmander L. (1994) Notions of Convexity. Boston: Birkhäuser.
[7] Singer I. (1997) Abstract Convex Analysis. New York: John Wiley & Sons.
[8] Pallaschke D. and Rolewicz S. (1998) Foundations of Mathematical Optimization, Convex Analysis Without Linearity. Dordrecht: Kluwer Academic Publishers.
[9] Rubinov A. M. (2000) Abstract Convexity and Global Optimization. Dordrecht: Kluwer Academic Publishers.
[10] Ioffe A. D. and Rubinov A. M. (2002) Abstract convexity and nonsmooth analysis. Global aspects. Adv. Math. Econom., 4, 1–23.
[11] Fenchel W. (1953) Convex Cones, Sets, and Functions. Princeton: Princeton Univ. Press.
[12] Kusraev A. G. and Kutateladze S. S. (2005) Introduction to Boolean-Valued Analysis. Moscow: Nauka Publishers [in Russian].
[13] 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.
[14] Kusraev A. G. and Kutateladze S. S. (2007) Subdifferential Calculus: Theory and Applications. Moscow: Nauka Publishers [in Russian].


This is an abstract of a talk prepared for the Conference on Nonlinear Analysis and Optimization, Technion (Haifa, Israel), June 18–24, 2008, in celebration of Alex Ioffe’s 70th and Simeon Reich’s 60th birthdays. The talk was undelivered due to illness.



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