2

INTRODUCTION

of measure was first extensively use d by Maxwell in his studies of statistical physic s

(Maxwell-Boltzmann distributio n law , 1866).

We wil l tak e a s th e poin t o f departur e fo r interaction s betwee n geometr y o f

high-dimensional structures , topologica l transformatio n groups , an d combinatoric s

the Dvoretzky theorem [49 , 50] , regarded i n light o f later result s by Vitali Milman .

As Aryeh Dvoretzky remarks in his 1959 Proc. Nat. Acad. Sci. USA note [49],

"this result , o r variou s weake r variations , hav e ofte n bee n con -

jectured. However , th e only explicit statemen t o f this conjectur e

in prin t know n t o th e autho r i s in a recen t pape r b y Alexandr e

Grothendieck" [97] .

Grothendieck's pape r i n question wa s published (i n the Bol. Soc. Mat. Sao Paulo)

when he was a visiting professor a t the Universidade de Sao Paulo, Brazil (1953-54),

and ca n b e als o regarded a s one o f the source s o f the presen t theory .

THEOREM

0.0. 1 (Dvoretzk y theorem , 1959). For every e 0 , each normed

space X of finite dimension n contains a subspace E of dimension k c(^)logn

;

which is Euclidean to within e:

dBM(EJ2{k))l

+ s.

Here c{e) — ce 2/|loge| for some absolute constant c 0.

In th e abov e result, dsM i s the multiplicative Banach-Mazur distance betwee n

two isomorphic norme d spaces :

dBM{E,F) = inf{||T| | •

^T~1]

\ : T i s an isomorphism}.

The geometri c meanin g o f the Banach-Mazu r distanc e i s clear fro m th e follow -

ing Fig. 0.1.

FIGURE

0.1. Banach-Mazu r distance : R

2

= ( 1 + e)R\.

A logarithmic dependenc e o f k on n is the best possibl e for al l sufficiently smal l

e 0 . Thus , generall y speaking , th e dimensio n o f a spac e X shoul d b e ver y hig h

in orde r t o guarante e th e existenc e o f "almos t elliptical " sections .

For instance , Fig . 0. 2 show s sections o f the uni t cub e l

n

(tha t is , the uni t bal l

of £°°(n)) b y rando m 2-planes . Thos e section s ar e normalize d t o th e sam e size .

Even i n dimensio n millio n (th e highes t th e autho r coul d manag e o n a standar d

desktop computer ) th e sectio n remain s rathe r rough . Thi s i s what theor y predicts ,

as the uni t cub e (tha t is , up t o a dilation, th e uni t bal l of £°°(n)) i s the wors t cas e

presently know n ([69] , Sect . 4.1).