January 10, Wednesday, 15:30, room 211

convex bodies inevitably have sections that are nearly ellipsoids.

Stated initially by Grothendieck as a conjecture in the asymptotic

theory of Banach spaces it was proved by Dvoretzky in 1961. But it was

only after Milman’s simple proof via isoperimetric inequality and

concentration of measure phenomenon (this is exactly the effect

responsible for a thick skin of a high-dimensional orange) that many

connections of this question with as diverse fields as Riemannian

geometry, functional analysis, combinatorics and computer science

became apparent.

This story can be viewed as a geometric manifestation of both the law

of large numbers of Probability theory and Ramsey theory of

Combinatorics. We shall discuss Milman’s proof of Dvoretzky-Milman

theorem.