More on embedding subspaces of L_{p} into ℓ_{p}^{N}, 0 < p < 1

Artem Zvavitch^{∗}

Abstract

It is shown that n-dimensional subspace of L_{p}, 0 < p < 1, (1 + ε)- embeds in l_{p}^{N}, whenever N ≥ c(ε, p)n(log n)(log log n)^{2}.

1 Introduction

Let X be a n-dimensional subspace of L_{p}, 0 < p < ∞, and let ε > 0. What is the smallest N = N(n, ε) such that there is a subspace Y of l_{p}^{N} for which d(X, Y) ≤ 1 + ε, where d(X, Y) is the Banach-Mazur distance?

In this paper we consider only the case 0 < p < 1. In this case, it is shown in [S-Z] that N(n, ε) grows in n at most like n(log n)^{3}. The tools used in [S-Z] are random choice argument, Lewis’ theorem in the non-convex setting and finally Dudley’s theorem for bounding of the supremum of a Gaussian process. Here instead of using Dudley’s theorem we will use the majorizing measures technique, developed by M. Talagrand.

M. Talagrand [T2] used this method to find an estimate for N(n, ε), for 1 < p < 2, which improved a previous estimate of [B-L-M] by reducing the power of the log n factor. We simplify a bit the methods used in [T2], and we improve the result of [S-Z] by proving the following theorem.

Theorem 1 Let 0 < p < 1 then

N(n, ε) ≤ C(p)ε^{−2}n(log n)(log log ε^{−2}n + log ε^{−1})^{2}.

∗supported by ISF

Throughout this paper C denotes absolute constant whose value may change from line to line, C(p), c(p) denote constants depending only on the argument p.

The proof is by the method of random selection of coordinates (for more details on this method see e.g. [J-S 2]). We may assume without loss of generality that X lies in a finite (but large) dimensional L_{p}^{M}(μ) space, where μ is a probability measure on {1, …, M}. We would like to show that the restriction operator to a set of relatively few of the M coordinates is a (1+ε)- isometry on X. We do it iteratively, reducing the size of M by a factor of at least 3/4 each time. To apply a random choice argument we first of all need to change the “location” of X in L_{p}^{M}(μ). This is done using the following version of Lewis’ theorem (see Theorem 2.1 in [S-Z] and Proposition 2.2(i) in [J-S 1]).

Theorem 2 Let X be an n dimensional subspace of L_{p}^{M}(μ), 0 < p < ∞. Then, one can find an integer K ≤ 3M/2, a probability measure λ on {1, …, K} and a subspace F of L_{p}(λ), isometric to X, which admits a basis

(h_{i})_{i≤n} orthogonal in L_{2}(λ) such that ∑_{i=1}^{n} h_{i}^{2} ≡ n. Moreover, for each i ≤ K

we have λ({i}) ≤ 2/K.

Next we will change the density in such a way that a new density ν will have the property 1/2K ≤ ν({i}) ≤ 2/K. We define

ν(\{i\}) = \frac{λ(\{i\}) + \frac{1}{K}}{2}

then ν is a probability measure on {1, …, K} and we can define an isometry I : L_{p}^{K}(λ) → L_{p}^{K}(ν) by

I(f)(i) = f(i)(\frac{λ(\{i\})}{ν(\{i\})})^{1/p}.

Define F_{1} = I(F), then it is shown in [S-Z], that there is K_{1} ≤ K/2 and a subspace Y of L_{p}^{K_{1}}(ν), such that d(F_{1}, Y) ≤ 1 + c(p)A_{F_{1}}, where

A_{F_{1}} = IE sup_{x∈F_{1},||x||_{p}≤1} | ∑_{i=1}^{K} ν({i})g_{i}|x(i)|^{p}|,

and the sequence {g_{i}}_{i=1}^{K} denote here a sequence of independent standard Gaussian variables. In Section 3 below, we will prove that

A_{F_{1}} ≤ C(p) (\frac{n}{K} log n [log \frac{K}{n} + log log K]^{2})^{\frac{1}{2}} .  (1)

Finally we can replace the bound given in Proposition 4.1 in [S-Z] by (1) and we follow Section 5 in [S-Z] to finish the proof of Theorem 1.

2 Entropy estimates.

Let B_{r}(X) = {x ∈ X : ||x||_{r} ≤ 1}. By N(B_{r}(X), tB_{s}(X)) we denote the metric entropy of B_{r}(X) with respect to tB_{s}(X), i.e., the minimal number of translates of tB_{s}(X) needed to cover B_{r}(X). We shell use the following result from [S-Z] (Proposition 3.1 there).

Lemma 1 Let 0 < p < 2, and let F be as in the Theorem 2. Then, for any t > 1,

log N(B_{p}(F), tB_{2}(F)) ≤ C(p)n(log n)t^{−2p/(2−p)}.

Now we would like to show that the same estimate (up to a constant) is valid for F_{1}.

||I(f)||_{2,ν}^{2} = ∑_{i=1}^{K} λ(\{i\})|f(i)|^{2}(\frac{2λ(\{i\})}{λ(\{i\}) + \frac{1}{K}} )^{2/p−1} ≤ 2^{2/p−1}||f||_{2,λ}^{2}

and then

N(B_{p}(F_{1}), tB_{2}(F_{1})) ≤ N(B_{p}(F), 2^{1/p−1/2}tB_{2}(F)),

consequently

N(B_{p}(F_{1}), 2^{1/p−1/2}tB_{2}(F_{1})) ≤ C(p)n(log n)t^{−2p/(2−p)}.  (2)

Let us define B_{p} = B_{p}(F_{1}) and B_{2} = B_{2}(F_{1}).

Lemma 2

log N(B_{p} + tB_{2}, 3tB_{2}) ≤ C(p)n(log n)t^{−2p/(2−p)}.  (3)

It is easy to see that N(B_{p}+tB_{2}, 3tB_{2}) ≤ log N(B_{p}, tB_{2}) and then the lemma follows from (2).

3 A Bound on the Gaussian process

As we explained in the introduction Theorem 1 will be proved once we establish the following theorem.

Theorem 3 Let 0 < p < 1, let g_{i}, i = 1, …, K, be independent standard Gaussian random variables and assume 1/2K ≤ ν({i}) ≤ 2/K. Then

IE sup_{x∈B_{p}} | ∑_{i=1}^{K} ν(\{i\})g_{i}|x(i)|^{p}| ≤ C(p) (\frac{n}{K} log n [log \frac{K}{n} + log log K]^{2})^{\frac{1}{2}} . (4)

To prove this theorem, we will use the majorizing measure bound (for details see [T2],[T3])

IE sup_{x∈B_{p}} | ∑_{i=1}^{K} ν({i})g_{i}|x(i)|^{p}| ≤ Cγ(B_{p}, d),  (5)

where

γ(B_{p}, d) = inf_{μ} sup_{x∈B_{p}} ∫_{0}^{∞} \sqrt{log \frac{1}{μ(B(x, ε))}}dε.  (6)

Here the infimum is taken over all probability measures μ on B_{p} and B(x, ε) is the d-ball where

d(y, z) = ( ∑_{i=1}^{K}[ν({i})(|y(i)|^{p} − |z(i)|^{p})]^{2})^{1/2}, y, z ∈ B_{p}.  (7)

Define

δ(y, z) = ( ∑_{i=1}^{K} ν^{2}({i})|y(i) − z(i)|^{2p})^{1/2}, y, z ∈ B_{p},  (8)

and let us show that δ is a metric. If p < 1/2 then 2p < 1 and

δ^{2}(y, z) ≤ δ^{2}(y, x) + δ^{2}(x, z),  (9)

which implies

δ(y, z) ≤ δ(y, x) + δ(x, z).  (10)

If 1 > p ≥ 1/2 then 2p ≥ 1 and

δ^{1/p}(y, z) ≤ δ^{1/p}(y, x) + δ^{1/p}(x, z),  (11)

which again implies

δ(y, z) ≤ δ(y, x) + δ(x, z).  (12)

Moreover, d(y, z) ≤ δ(y, z) for any y, z ∈ B_{p}, which clearly implies that γ(B_{p}, d) ≤ γ(B_{p}, δ). Our goal now is to bound γ(B_{p}, δ).

Consider a metric space (T, ρ) and fix an integer s ≥ 1. Following Talagrand [T2] we next define the notion of an s-tree .

Recall that a tree of subsets of T is a collection F of subsets of T with the property that for all A, B ∈ F, either A ∩ B = ∅, or A ⊂ B, or B ⊂ A. We say that B is a son of A if B ⊂ A and

C ∈ F, B ⊂ C ⊂ A ⟹ C = B or C = A.  (13)

A branch of F is a sequence A_{1} ⊃ A_{2} ⊃ … such that A_{k+1} is a son for A_{k}. A branch is maximal if it is not contained in a longer branch. To each A ∈ F we denote by N(A) the number of sons of A. Set r = 8. A tree F called an s-tree if we can associate to each A ∈ F an integer n(A) in manner that

1. diam(A) ≤ r^{−n(A)}.

2. If B ⊂ A then

n(B) ≥ n(A) + s.  (14)

3. If B, B′ are two distinct sons of A then

ρ(B, B′) ≥ \frac{1}{2}r^{−n(A)−1}.  (15)

The size γ(F, ρ) of F is the infimum over all possible maximal branches of

∑_{k≥1} r^{−n(A_{k})}\sqrt{log N(A_{k})}.  (16)

Theorem 4 ( Talagrand [T2]) For some universal constant C and for s ≥ 1, any metric space (T, ρ), with γ(T, ρ) < ∞, contains an s-tree satisfying

γ(F, ρ) ≥ \frac{1}{Cs}γ(T, ρ).  (17)

Consider a parameter s to be determinated later. We would like to bound the size of an s-tree F contained in (B_{p}, δ) and show that

γ(F, δ) ≤ \sqrt{\frac{n log n}{K}},  (18)

for

s ≤ C(p) [log \frac{K}{n} + log log K] ,  (19)

thus establish (4).

Let n_{1} to be the smallest integer such that for some A ∈ F we have n(A) = n_{1}. We pick A_{1} such that n(A_{1}) = n_{1}. Starting with A_{1} we will construct a branch (A_{k})_{k≥1} of F, disjoint subsets I_{k} of J = {1, …, K} and non negative numbers (a_{k})_{k≥1} such that

∀y ∈ A_{k+1}, ∑_{i∈I_{k}} ν({i})|y(i)|^{p} ≥ a_{k},  (20)

and

r^{−n_{k}} \sqrt{log N_{k}} ≤ min \{C(p)(\frac{1}{log K} + a_{k})\sqrt{\frac{n log n}{K}}, C(p)r^{−n_{k}} \sqrt{n}\} , (21)

where n_{k} = n(A_{k}), N_{k} = N(A_{k}). Observe that necessarily ∑ a_{k} ≤ 1. Once (21) is established, to bound γ(F, δ), we simply use that by definition

γ(F, δ) ≤ ∑_{k≥1} r^{−n_{k}} \sqrt{log N_{k}}.  (22)

Consider the smallest integer k_{0} with N_{k_{0}} ≥ 2. Fix some k_{1} > k_{0} then

∑_{k≥1} r^{−n_{k}} \sqrt{log N_{k}} = ∑_{k≥k_{0}} r^{−n_{k}} \sqrt{log N_{k}} ≤

C(p)(\frac{k_{1} − k_{0}}{log K} + 1)\sqrt{\frac{n log n}{K}} + C(p)r^{−n_{k_{1}}} \sqrt{n}.

It is easy to see that the δ-diameter of B_{p} is at most 4. Since N_{k_{0}} ≥ 2, n_{k_{0}} > 1; thus, since n_{k_{1}} ≥ n_{k_{0}} + (k_{1} − k_{0})s, if k_{1} − k_{0} = log K, we have r^{−n_{k_{1}}} ≤ 1/\sqrt{K}, so that in particular, we see that

γ(F, δ) ≤ C(p)\sqrt{\frac{n log n}{K}}.  (23)

We now turn to the construction. To make the construction possible, we will also require that

ν(I_{k}) ≤ (c̄K)^{−1}r^{2n_{k}},  (24)

where c̄ = c̄(p) ∈ (0, 1] is a constant to be chosen later. We set I_{0} = ∅. Assuming that A_{k}, I_{k−1} have been constructed, we construct A_{k+1}, I_{k}.

Using standard volume considerations in the n-dimensional space (F_{1}, || ∗ ||_{2p,ν^{2}}), we observe that log N_{k} ≤ C(p)n so that

r^{−n_{k}} \sqrt{log N_{k}} ≤ C(p)r^{−n_{k}} \sqrt{n},  (25)

and the second term in (21) does not cause any problem. Case 1 If

r^{−n_{k}} \sqrt{log N_{k}} ≤ \frac{1}{log K} \sqrt{\frac{n log n}{K}},  (26)

then set I_{k} = ∅, a_{k} = 0, and we chose A_{k+1} to be any of the sons of A_{k}. Case 2 We have

r^{−n_{k}} \sqrt{log N_{k}} ≥ \frac{1}{log K} \sqrt{\frac{n log n}{K}},  (27)

(i.e the number of sons is not too small). We denote as (X_{l})_{l≤N_{k}} the sons of A_{k}. For each l ≤ N_{k}, we pick x_{l} ∈ X_{l}. We

set J_{k} = ⋃_{l≤k−1} I_{l}, then

ν(J_{k}) ≤ 2(c̄K)^{−1}r^{2n_{k−1}}.  (28)

We set η_{k} = c̄Kr^{−2n_{k}} and we consider

S = sup_{l≤N_{k}} ∑_{J\J_{k}} ν({i})|x_{l}(i)|^{p}1_{{|x_{l}(i)|^{p}≥η_{k}}}.  (29)

The Main Claim in case 2 is that

c(p)\frac{r^{−n_{k}} \sqrt{log N_{k}}}{\sqrt{\frac{n log n}{K}}} ≤ S.  (30)

Before proving the “Main Claim”, we show how to conclude the construction. We select l ≤ N_{k} that achieves the sup in S. We set A_{k+1} = X_{l} and

I_{k} = {i ≤ K; i ∉ J_{k}; |x_{l}(i)|^{p} ≥ η_{k}}.  (31)

Note that

ν(I_{k}) ≤ η_{k}^{−1} = (c̄K)^{−1}r^{2n_{k}}.

We also set a_{k} = S/2. Consider y ∈ A_{k+1} (we need to prove that

∑_{i∈I_{k}} ν({i})|y(i)|^{p} ≥ a_{k}). Then

δ(y, x_{l}) ≤ r^{−n_{k+1}} ≤ r^{−n_{k}−s}  (32)

since F is an s-tree. Thus

∑_{i∈I_{k}} ν({i})||y(i)|^{p} − |x_{l}(i)|^{p}| ≤ ∑_{i∈I_{k}} ν({i})|y(i) − x_{l}(i)|^{p} ≤  (33)

(by Cauchy-Schwartz)

( ∑_{i∈I_{k}} ν^{2}({i})|y(i) − x_{l}(i)|^{2p})^{1/2} (card{i : i ∈ I_{k}})^{1/2} ≤  (34)

(by ν(\{i\}) ≥ \frac{1}{2k} and ν(I_{k}) ≤ (c̄K)^{−1}r^{2n_{k}})

( ∑_{i∈I_{k}} ν^{2}({i})|y(i) − x_{l}(i)|^{2p})^{1/2}(2Kν(I_{k}))^{1/2} ≤ C(p)r^{−s}.  (35)

Now by (27) and the “Main Claim” we have

S ≥ c(p)\frac{1}{log K} ,  (36)

and we get

∑_{i∈I_{k}} ν(\{i\})||y(i)|^{p} − |x_{l}(i)|^{p}| ≤ \frac{S}{2} ,  (37)

as soon as

r^{s} ≥ C(p)(log K),  (38)

i.e. s ≥ C(p) log log K. Finally,

∑_{i∈I_{k}} ν(\{i\})|y(i)|^{p} ≥ \frac{S}{2} = a_{k}.  (39)

Proof of the “Main Claim”: We have that

∀l, l′ ≤ N_{k}, l ≠ l′, \frac{r^{−n_{k}−1}}{2} ≤ δ(x_{l}, x_{l′}) ≤ r^{−n_{k}},  (40)

define X = {x_{l}}_{l≤N_{k}}.

Proposition 1 If r^{2s} ≥ C(p)\frac{K log^{2} K}{n log n} ( s ≥ C(p) (log \frac{K}{n} + log log K)), then we can find a subset L of {1, …, N_{k}} with

log cardL ≥ \frac{1}{2} log N_{k}  (41)

and such that for all l, l′ ∈ L, l ≠ l′:

∑_{i∈J\J_{k}} ν^{2}(\{i\})|x_{l}(i) − x_{l′}(i)|^{2p} ≥ \frac{r^{−2n_{k}−2}}{8} .  (42)

Proof : Consider

H = \{x ∈ B_{p} : ∑_{i∈J_{k}} ν^{2}(\{i\})|x(i)|^{2p} ≤ \frac{r^{−2n_{k}−2}}{32} \}.

Fix l_{0} ≤ N_{k}, from δ^{2}(x_{l}, x_{l_{0}}) ≤ \frac{r^{−2n_{k}−2}}{32} 32r^{2}, we get that

x_{l} − x_{l_{0}} ∈ (32r^{2})^{1/2p}H, for all l ≤ N_{k},

i.e. X ⊂ x_{l_{0}} + (32r^{2})^{1/2p}H. By standard volume considerations we get

log N(X , H) ≤ log((8r^{2})^{1/2p}H, H)

≤ C(p)cardJ_{k} ≤ C(p)r^{2n_{k−1}} ≤ C(p)r^{−2s}r^{2n_{k}} ≤ (by inequality (27) and bound on s from the statement of the proposition)

C(p)r^{−2s} \frac{K log^{2} K}{n log n} log N_{k} ≤ \frac{1}{2} log N_{k},

i.e. log N(X , H) ≤ \frac{1}{2} log N_{k} or N(X , H) ≤ \sqrt{N_{k}}.

Next we use the “pigeon hole principle”: we can cover X by Q translates of the set H where Q ≤ \sqrt{N}_{k}. One of these sets must contain at least \sqrt{N_{k}} points of X, let L to be the set of indexes of these points. Finally, applying the triangle inequality we get that

∑_{i∈J_{k}} ν^{2}(\{i\})|x_{l}(i) − x_{l′}(i)|^{2p} ≤ \frac{r^{−2n_{k}−2}}{8} ,  (43)

for all l, l′ ∈ L, l ≠ l′, and (42) easily follows from (43) and (40).

□

Proposition 2 Let L satisfy the conditions of Proposition 1 and l, l′ ∈ L, l ≠ l′. Then

||x_{l} − x_{l′}||_{2}^{2p/(2−p)} ≥ C \frac{Kr^{−2n_{k}}}{S^{q}} ,  (44)

where q = 2(1 − p)/(2 − p).

Proof : Fix l, l′ ∈ L, l ≠ l′, then

∑_{J\J_{k}} ν^{2}({i})|x_{l}(i) − x_{l′}(i)|^{2p}1_{{|x_{l}(i)|^{p}≤η_{k},|x_{l′}(i)|^{p}≤η_{k}}} ≤  (45)

(by ν(\{i\}) ≤ \frac{2}{K} and |x_{l}(i) − x_{l′}(i)|^{p} ≤ |x_{l}(i)|^{p} + |x_{l′}(i)|^{p})

\frac{2}{K} 2η_{k} ∑_{J\J_{k}} ν(\{i\})|x_{l}(i) − x_{l′}(i)|^{p} ≤ \frac{8(c̄Kr^{−2n_{k}})}{K} ≤ \frac{r^{−2n_{k}−2}}{16} ,  (46)

as soon as c̄ ≤ (128r^{2})^{−1}. By the properties of L,

∑_{i∈J\J_{k}} ν^{2}(\{i\})|x_{l}(i) − x_{l′}(i)|^{2p}1_{\{|x_{l}(i)|^{p}>η_{k} or |x_{l′}(i)|^{p}>η_{k}\}} ≥ \frac{r^{−2n_{k}−2}}{16} . (47)

On the other hand Hölder inequality implies (see e.g. [S-Z] proof of Proposition 4.1),

∑_{i∈J\J_{k}} ν^{2}({i})|x_{l}(i) − x_{l′}(i)|^{2p}1_{{|x_{l}(i)|^{p}>η_{k} or |x_{l′}(i)|^{p}>η_{k}}} ≤

6K^{−1}||x_{l} − x_{l′}||2^{\frac{2p}{2−p}} ( ∑_{i∈J\J_{k}} ν(\{i\})|x_{l}(i) − x_{l′}(i)|^{p}1_{\{|x_{l}(i)|^{p}>η_{k} or |x_{l′}(i)|^{p}>η_{k}\}})^{q} .

But |x_{l}(i) − x_{l′}(i)|^{p} ≤ |x_{l}(i)|^{p} + |x_{l′}(i)|^{p}, and it is easy to see that

(|x_{l}(i)|^{p} + |x_{l′}(i)|^{p}) 1_{{|x_{l}(i)|^{p}>η_{k} or |x_{l′}(i)|^{p}>η_{k}}}
≤ 2(|x_{l}(i)|^{p}1_{{|x_{l}(i)|^{p}≥η_{k}}} + |x_{l′}(i)|^{p}1_{{|x_{l′}(i)|^{p}≥η_{k}}}),  (48)

and by the definition of S (29),

∑_{i∈J\J_{k}} ν({i})|x_{l}(i) − x_{l′}(i)|^{p}1_{{|x_{l}(i)|^{p}>η_{k} or |x_{l′}(i)|^{p}>η_{k}}} ≤ 4S.  (49)
□

Fix some l_{0} ∈ L and consider the points z_{l} = x_{l} − x_{l_{0}}. Then z_{l} ∈ 2^{1/p}B_{p} and

||z_{l} − z_{l′}||_{2}^{2p/(2−p)} ≥ C \frac{Kr^{−2n_{k}}}{S^{q}} .  (50)

Applying the entropy estimate from Lemma 1 we get

log N_{k} ≤ C(p)\frac{n log n}{K}r^{2n_{k}}S^{q},  (51)

or

c(p)\frac{r^{−n_{k}} \sqrt{log N_{k}}}{\sqrt{\frac{n log n}{K}}} ≤ S^{q/2},  (52)

This estimation is weaker then the “Main Claim” (because q/2 = \frac{1−p}{2−p} < 1 ), however, from (27) and (51) we can get the following very useful lower bound on S:

\frac{c(p)}{log K} ≤ S^{\frac{1−p}{2−p}} ,  (53)

and finally

(\frac{c(p)}{log K} )^{\frac{2−p}{1−p}} ≤ S.  (54)

Proposition 3 If r^{s} ≥ (\frac{log K}{c(p)} )^{\frac{2−p}{1−p}} (s ≥ C(p) log log K), then

∑_{i∈J_{k}} ν({i})|z_{l}(i)|^{p} ≤ S,  (55)

for every l ∈ L.

Proof :

∑_{i∈J_{k}} ν({i})|x_{l}(i) − x_{l_{0}}(i)|^{p} ≤ ( ∑_{i∈J_{k}} ν^{2}({i})|x_{l}(i) − x_{l_{0}}(i)|^{2p})^{1/2}(2Kν(J_{k}))^{1/2} ≤

C(p)r^{−n_{k}+n_{k−1}} ≤ C(p)r^{−s} ≤ (\frac{log K}{c(p)} )^{− \frac{2−p}{1−p}} ≤ S.  (56)

□

We are now ready to conclude the proof of the “Main Claim”. For l ∈ L we represent z_{l} as

{z_{l}(i)}_{i=1}^{K} = {z_{l}(i)}_{i∈J_{k}} + {z_{l}(i)}_{i∉J_{k},|z_{l}(i)|^{p}≥2η_{k}} + {z_{l}(i)}_{i∉J_{k},|z_{l}(i)|^{p}<2η_{k}}. (57)

Applying Proposition 3 we get {z_{l}(i)}_{i∈J_{k}} ∈ S^{1/p}B_{p} and from (48) and the definition of S (29) we get that

{z_{l}(i)}_{i∉J_{k},|z_{l}(i)|^{p}≥2η_{k}} ∈ (4S)^{1/p}B_{p}.  (58)

Using ||x||_{2}^{2} ≤ ||x||_{∞}^{2−p}||x||_{p}^{p}, we get

\{z_{l}(i)\}_{i∉J_{k},|z_{l}(i)|^{p}<2η_{k}} ∈ 2(2η_{k})^{\frac{2−p}{2p}} B_{2}.  (59)

Combining, we get

z_{l} ∈ C(p) (S^{1/p}B_{p} + ηk^{\frac{2−p}{2p}} B_{2}) .  (60)

By (50) and (60),

log N_{k} ≤ C log N(S^{1/p}B_{p} + ηk^{\frac{2−p}{2p}} B_{2}, c(p)\frac{(Kr^{−2n_{k}})^{\frac{2−p}{2p}}}{S^{\frac{1−p}{p}}} B_{2})

= C log N(B_{p} + (η_{k})^{\frac{2−p}{2p}} S^{−1/p}B_{2}, c(p)\frac{(Kr^{−2n_{k}})^{\frac{2−p}{2p}}}{S^{\frac{2−p}{p}}} B_{2}). (61)

To improve upon (51) we want to find a good bound for (61), which is done by applying Lemma 2. For that we need

3(η_{k})^{\frac{2−p}{2p}} S^{−1/p} ≤ c(p)\frac{(Kr^{−2n_{k}})^{\frac{2−p}{2p}}}{S^{\frac{2−p}{p}}} ,  (62)

but, by definition, η_{k} = c̄Kr^{−2n_{k}} and S ≤ 1, so for (62) to be true we just have to take c̄ ≤ c(p). Finally, we apply Lemma 2 and get

log N_{k} ≤ C(p)\frac{n log n}{K}r^{2n_{k}}S^{2},  (63)

which establish the “Main Claim”(30).

□

References

[B-L-M] J. Bourgain, J. Lindenstrauss and V. Milman (1989)Approximation of zonoids by zonotopes, Acta, Math, 162 73-141.

[J-S 1] W.B. Johnson and G. Schechtman (1994) Computing p-Summing Norms With Few Vectors, Israel Journal of Math. 87, 19-31.

[J-S 2] W.B. Johnson and G. Schechtman Finite dimensional L_{p} spaces, 1 ≤ p < ∞, in preparation.

[S-Z] G. Schechtman, A. Zvavitch Embedding subspaces of L_{p} into ℓ_{p}^{N}, 0 < p < 1, to appear in Math. Nachr.

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[T2] M. Talagrand Embedding subspaces of L_{p} in L_{p}^{N}. Geometric aspects of functional analysis (Israel, 1992-1994), 311-325, Oper. Theory Adv. Appl., 77, Birkhuser, Basel, 1995.

[T3] M. Talagrand Majorizing measures: the generic chaining.(1996), Ann. of Probability. 24, no. 3, 1049-1103.