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