Tiểu Luận Metric cotype

Metric cotype
ByManor MendelandAssaf Naor
Abstract
We introduce the notion ofcotype of a metric space, and prove that for
Banach spaces it coincides with the classical notion of Rademacher cotype.
This yields a concrete version of Ribe’s theorem, settling a long standing open
problem in the nonlinear theory of Banach spaces. We apply our results to
several problems in metric geometry. Namely, we use metric cotype in the
study of uniform and coarse embeddings, settling in particular the problem
of classifying whenLpcoarsely or uniformly embeds intoLq. We also prove a
nonlinear analog of the Maurey-Pisier theorem, and use it to answer a question
posed by Arora, Lov´asz, Newman, Rabani, Rabinovich and Vempala, and to
obtain quantitative bounds in a metric Ramsey theorem due to Matouˇsek.
1. Introduction
In 1976 Ribe [62] (see also [63], [27], [9], [6]) proved that if XandY
are uniformly homeomorphic Banach spaces thenXis finitely representable in
Y, and vice versa (Xis said to be finitely representable in Yif there exists a
constantK>0 such that any finite dimensional subspace ofXisK-isomorphic
to a subspace ofY). This theorem suggests that “local properties” of Banach
spaces, i.e. properties whose definition involves statements about finitely many
vectors, have a purely metric characterization. Finding explicit manifestations
of this phenomenon for specific local properties of Banach spaces (such as type,
cotype and super-reflexivity), has long been a major driving force in the bi-Lipschitz theory of metric spaces (see Bourgain’s paper [8] for a discussion
of this research program). Indeed, as will become clear below, the search
for concrete versions of Ribe’s theorem has fueled some of the field’s most
important achievements.
The notions of type and cotype of Banach spaces are the basis of a deep and
rich theory which encompasses diverse aspects of the local theory of Banach
spaces. We refer to [50], [59], [58], [68], [60], [36], [15], [71], [45] and the
references therein for background on these topics. A Banach spaceXis said
248 MANOR MENDEL AND ASSAF NAOR
to have (Rademacher) typep>0 if there exists a constantT<∞such that
for every nand everyx1, . ,xn∈X,
Eε

n 
j=1
εjxj

p
X
≤T
p
n 
j=1
xj
p
X. (1)
where the expectationEε is with respect to a uniform choice of signs ε =
(ε1, . ,εn)∈{ư1,1}
n
. Xis said to have (Rademacher) cotype q>0 if there
exists a constantC<∞such that for everynand everyx1, . ,xn∈X,
Eε

n 
j=1
εjxj

q
X
≥
1
Cq
n 
j=1
xj
q
X. (2)
These notions are clearlylinear notions, since their definition involves ad-dition and multiplication by scalars. Ribe’s theorem implies that these notions
are preserved under uniform homeomorphisms of Banach spaces, and therefore
it would be desirable to reformulate them using only distances between points
in the given Banach space. Once this is achieved, one could define the no-tion of type and cotype of a metric space, and then hopefully transfer some of
the deep theory of type and cotype to the context of arbitrary metric spaces.
The need for such a theory has recently received renewed impetus due to the
discovery of striking applications of metric geometry to theoretical computer
science (see [44], [28], [41] and the references therein for part of the recent
developments in this direction).
Enflo’s pioneering work [18], [19], [20], [21] resulted in the formulation
of a nonlinear notion of type, known today asEnflo type. The basic idea is
that given a Banach spaceXandx1, . ,xn∈X, one can consider the linear
function f:{ư1,1}
n→Xgiven byf(ε)=
n
j=1εjxj. Then (1) becomes
(3) Eεf(ε)ưf(ưε)
p
X≤T
p
n 
j=1
Eε

f(ε1, . ,εjư1,εj,εj+1, . ,εn)
ưf(ε1, . ,εjư1,ưεj,εj+1, . ,εn)

p
X
.
One can thus say that a metric space (M,dM) has Enflo typepif there exists
a constantTsuch that for everyn∈Nandeveryf:{ư1,1}
n→M,
(4) EεdM(f(ε),f(ưε))
p
≤T
p
n 
j=1
EεdM

f(ε1, . ,εjư1,εj,εj+1, . ,εn),
f(ε1, . ,εjư1,ưεj,εj+1, . ,εn)
p
.
There are two natural concerns about this definition. First of all, while in
the category of Banach spaces (4) is clearly a strengthening of (3) (as we
are not restricting only to linear functionsf), it isn’t clear whether (4) follows
METRIC COTYPE 249
from (3). Indeed, this problem was posed by Enflo in [21], and in full generality
it remains open. Secondly, we do not know if (4) is a useful notion, in the
sense that it yields metric variants of certain theorems from the linear theory
of type (it should be remarked here that Enflo found striking applications of
his notion of type to Hilbert’s fifth problem in infinite dimensions [19], [20],
[21], and to the uniform classification of Lp spaces [18]). As we will presently
see, in a certain sense both of these issues turned out not to be problematic.
Variants of Enflo type were studied by Gromov [24] and Bourgain, Milman
and Wolfson [11]. Following [11] we shall say that a metric space (M,dM) has
BMW typep>0 if there exists a constantK<∞such that for everyn∈N
and everyf:{ư1,1}
n→M,
(5)
EεdM(f(ε),f(ưε))
2
≤K2
n
2
pư1
n 
j=1
EεdM

f(ε1, . ,εjư1,εj,εj+1, . ,εn),
f(ε1, . ,εjư1,ưεj,εj+1, . ,εn)
2
.
Bourgain, Milman and Wolfson proved in [11] that if a Banach space has
BMW typep>0 then it also has Rademacher typep

for all 0<p
<p. They
also obtained a nonlinear version of the Maurey-Pisier theorem for type [55],
[46], yielding a characterization of metric spaces which contain bi-Lipschitz
copies of the Hamming cube. In [59] Pisier proved that for Banach spaces,
Rademacher typepimplies Enflo type p

for every 0 <p
<p. Variants of
these problems were studied by Naor and Schechtman in [53]. A stronger
notion of nonlinear type, known as Markov type, was introduced by Ball [4] in
his study of theLipschitz extension problem. This important notion has since
found applications to various fundamental problems in metric geometry [51],
[42], [5], [52], [48]
Despite the vast amount of research on nonlinear type, a nonlinear notion
of cotype remained elusive. Indeed, the problem of finding a notion of cotype
which makes sense for arbitrary metric spaces, and which coincides (or almost
coincides) with the notion of Rademacher type when restricted to Banach
spaces, became a central open problem in the field.
There are several difficulties involved in defining nonlinear cotype. First
of all, one cannot simply reverse inequalities (4) and (5), since the resulting
condition fails to hold true even for Hilbert space (withp= 2). Secondly, if
Hilbert space satisfies an inequality such as (4), then it must satisfy the same
inequality where the distances are raised to any power 0 <r <p. This is
because Hilbert space, equipped with the metricxưy
r/p
, is isometric to a
subset of Hilbert space (see [65], [70]). In the context of nonlinear type, this
observation makes perfect sense, since if a Banach space has typepthen it
also has typerfor every 0 <r <p. But, this is no longer true for cotype