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).
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