To Vex
(WordWeb 5.0) 1.
Cause annoyance in; disturb, especially by minor irritations
2.
Disturb the peace of mind of; afflict with mental agitation or distress
3.
Change the arrangement or position of
4.
Subject to prolonged examination, discussion, or deliberation—“vex the subject of the death penalty”
5.
Be a mystery or bewildering to—“a vexing problem”
This is an overview of the origin, evolution,
and basics of convexity.
Study of convexity in the Sobolev Institute was initiated by
Leonid Kantorovich (1912–1986) and
Alexandr Alexandrov (1912–1999).
This talk is part of their memory.
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.
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[2] Ferrari G. R. F. (Ed.) (2000)
Plato: The Republic.
Berkeley: University of California (Translated by Tom Griffith).
[3] Macaulay G. C. (Tr.) (1890)
The History of Herodotus.
London and New York: Macmillan.
[4] Datta B. (1993)
Ancient Hindu Geometry: The Science of the Sulba.
New Delhi: Cosmo Publishers.
[5] Seidenberg A. (1978)
The origin of mathematics.
Archive for History of Exact Sciences.
18, 301-342.
[6] 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.
[7] Kutateladze S. S. and Rubinov A. M. (1972)
Minkowski Duality and Its Applications.
Russian Math. Surveys, 27:3, 137–191.
[8] Kutateladze S. S. and Rubinov A. M. (1976)
Minkowski Duality and Its Applications.
Novosibirsk: Nauka Publishers [in Russian].
[9] Fuchssteiner B. and Lusky W. (1981)
Convex Cones. Amsterdam: North-Holland.
[10] Hörmander L. (1994)
Notions of Convexity. Boston: Birkhäuser.
[11] Singer I. (1997)
Abstract Convex Analysis.
New York: John Wiley & Sons.
[12] Pallaschke D. and Rolewicz S. (1998)
Foundations of Mathematical Optimization, Convex
Analysis Without Linearity. Dordrecht: Kluwer Academic Publishers.
[13] Rubinov A. M. (2000)
Abstract Convexity and Global Optimization. Dordrecht:
Kluwer Academic Publishers.
[14] Ioffe A. D. and Rubinov A. M. (2002)
Abstract convexity and nonsmooth analysis. Global aspects.
Adv. Math. Econom., 4, 1–23.
[15] Fenchel W. (1953)
Convex Cones, Sets, and Functions.
Princeton: Princeton Univ. Press.
[16] Hörmander L. (1955)
Sur la fonction d'appui des ensembles convexes dans une espace lokalement
convexe. Ark. Mat., 3:2, 180–186 [in French].
[17] Litvinov G. L., Maslov V. P., and Shpiz G. B. (2001)
Idempotent functional analysis: an algebraic approach.
Math. Notes, 69:5, 758–797.
[18] Cohen G., Gaubert S., and Quadrat J.-P. (2002)
Duality and Separation Theorems in Idempotent Semimodules.
Rapporte de recherche No. 4668, Le Chesnay CEDEX: INRIA Rocquetcourt. 26 p.
[19] Kusraev A. G. and Kutateladze S. S. (2005)
Introduction to Boolean-Valued Analysis.
Moscow: Nauka Publishers [in Russian].
[20] 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.
[21] Kusraev A. G. and Kutateladze S. S. (2007)
Subdifferential Calculus: Theory and Applications.
Moscow: Nauka Publishers [in Russian].
This talk was delivered on September 20, 2007,
at the conference
“Mathematics in the Modern World”
dedicated to the fiftieth anniversary of the Sobolev Institute.