interaction of order and convexity s.s. kutateladze
TRANSCRIPT
Interaction ofInteraction ofOrder and ConvexityOrder and Convexity
S.S. Kutateladze
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.
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.’’
His ContributionHis Contribution
Abstract Abstract ConvexityConvexity
H-ConvexityH-Convexity
Hyperspaces Hyperspaces
Reshetnyak’sReshetnyak’sTheoremTheorem
ChoquetChoquet Theory Theory
MajorizationMajorization
Alexandrov’sAlexandrov’s Theorem Theorem
Dual Dual HyperspaceHyperspace
IdentificationIdentification
Blaschke’s SumBlaschke’s Sum
Mixed Volume Mixed Volume as Dualityas Duality
Dual ConesDual Cones
Decomposition Decomposition and Inclusionand Inclusion
Isoperimetric-Type Isoperimetric-Type ProblemsProblems
Problem’sProblem’sIngredientsIngredients
Convex Convex ProblemsProblems
SoapSoap Bubbles Bubbles
ExternalExternal Problems Problems
Uryhson-TypeUryhson-TypeProblemsProblems
CurrentCurrent Hyperplanes Hyperplanes