Interaction of Order and Convexity
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Transcript of Interaction of Order and Convexity
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Interaction ofInteraction ofOrder and ConvexityOrder and Convexity
S.S. Kutateladze
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The Gift of MathematicsThe Gift of Mathematics
Mathematics is not a divine gift. Mathematics is a human enterprise, challenge, and endeavor. The gift of mathematics goes from master to student. The alternating chain of masters and students is the true savior of mathematics.
Geometry is a vast treasure trove founded by Euclid and enriched by his innumerous students and followers. Alexandr Alexandrov was one of them.
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AlexandrovAlexandrov (1912-1999)(1912-1999)
Alexandr Danilovich Alexandrov was the first and foremost Russian geometer of the twentieth century.
He contributed to mathematics under the slogan:
“Retreat to Euclid.’’
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His ContributionHis Contribution
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Abstract Abstract ConvexityConvexity
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H-ConvexityH-Convexity
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Hyperspaces Hyperspaces
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Reshetnyak’sReshetnyak’sTheoremTheorem
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ChoquetChoquet Theory Theory
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MajorizationMajorization
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Alexandrov’sAlexandrov’s Theorem Theorem
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Dual Dual HyperspaceHyperspace
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IdentificationIdentification
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Blaschke’s SumBlaschke’s Sum
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Mixed Volume Mixed Volume as Dualityas Duality
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Dual ConesDual Cones
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Decomposition Decomposition and Inclusionand Inclusion
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Isoperimetric-Type Isoperimetric-Type ProblemsProblems
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Problem’sProblem’sIngredientsIngredients
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Convex Convex ProblemsProblems
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SoapSoap Bubbles Bubbles
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ExternalExternal Problems Problems
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Uryhson-TypeUryhson-TypeProblemsProblems
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CurrentCurrent Hyperplanes Hyperplanes