An extension of Milman's reverse Brunn-Minkowski inequality

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by  Jesus Bastero

An extension of Milman’s reverse Brunn-Minkowski inequality
by
Jesu´s Bastero *, Julio Bernu´es * and Ana Pen˜a *
Departamento deMatema´ticas. FacultaddeCiencias
UniversidaddeZaragoza
50009-Zaragoza (Spain)
5
9
9
1 0. Introduction

n
a The classical Brunn-Minkowski inequality states that for A1,A2 ⊂IRn compact,
J
4 |A1+A2|1/n ≥|A1|1/n+|A2|1/n (1)

] where |·| denotes the Lebesgue measure on IRn. Brunn[Br] gavethe firstproofof this inequality for
A
A ,A compact convex sets, followed by an analytical proofby Minkowski[Min]. The inequality (1)
1 2
F for compact sets, not necessarily convex, was first proved by Lusternik [Lu]. A very simple proof of
h. it can be found in [Pi 1], Ch. 1.
t Itiseasytoseethatonecannotexpectthereverseinequalitytoholdatall,evenifitisperturbed
a
m by a fixed constant and we restrict ourselves to balls (i.e. convex symmetric compact sets with the
origin as an interior point). Take for instance A = {(x ...x ) ∈ IRn | |x | ≤ ε,|x | ≤ 1,2 ≤ i ≤ n}
[ 1 1 n 1 i
and A ={(x ...x )∈IRn ||x |≤ε,|x |≤1,1≤i≤n−1}.
2 1 n n i
1
In1986V.Milman[Mil1]discoveredthatifB andB areballsthereisalwaysarelativeposition
v 1 2
0 ofB1andB2 forwhichaperturbedinverseof(1)holds. Moreprecisely:“ThereexistsaconstantC >0
1 such that for all n∈IN and any balls B ,B ⊂IRn we can find a linear transformation u:IRn →IRn
1 2
2 with |det(u)|=1 and
1
0
5 |u(B1)+B2|1/n ≤C(|B1|1/n+|B2|1/n)”
9
/
h
The nature of this reverse Brunn-Minkowski inequality is absolutely different from others (say
t
a reverse Blaschke-Santalo´inequality, etc.). Brunn-Minkowskiinequality is an isoperimetric inequality,
m (inIRn itis its firstandmostimportantconsequencetillnow)andthereis noinversetoisoperimetric
: inequalities. So, it was a new idea that in the class of affine images of convex bodies there is some
v
i kind of inverse.
X
The resultprovedby Milmanusedhardtechnicaltools(see[Mil 1]). Pisierin[Pi 2] gaveanew
r
a proof by using interpolation and entropy estimates. Milman in [Mil 2] gave another proof by using
the “convex surgery” and achieving also some entropy estimates.
The aimofthis paper is to extend this Milman’s resultto a largerclass ofsets. Note thatsimple
examples show that some conditions on a class of sets are clearly necessary.
For B ⊂IRn body (i.e. compact,with non empty interior),considerB =B−x , wherex is an
1 0 0
interiorpoint. IfwedenotebyN(B )=∩ aB thebalancedkernelofB ,itisclearthatN(B )is
1 |a|≥1 1 1 1
a balanced compact neighbourhood of the origin, so there exists c>0 such that B +B ⊂cN(B ).
1 1 1
TheAoki-Rolewicztheorem(see[Ro],[K-P-R])impliesthatthereis0<p≤1,namelyp=log−1(c),
2
such that B ⊂B¯ ⊂21/pB , where B¯ is the unit ball of some p-norm. This observation will allow us
1 1
to work in a p-convex enviroment.
The above construction allows us to define the following parameter. For B a body let p(B), 0<
p(B)≤1, be the supremum of the p for which there exist a measure preserving affine transformation
* Partially supported by Grant DGICYT PS 90-0120
AMS Class.: 46B20, . Key words: p-convexity, entropy
1
of B, T(B), and a p-norm with unit ball B¯ verifying T(B) ⊂ B¯ and |B¯| ≤ |81/pB|, (by suitably
adapting the results appearing in [Mil 2], it is clear that p(B)≥p for any p-convex body B).
Our main theorem is,
Theorem 1. Let0<p≤1. ThereexistsC =C(p)≥1suchthatforalln∈IN andallA ,A ⊂IRn
1 2
bodies such that p(A ),p(A ) ≥ p, there exists an affine transformation T(x) = u(x) + x with
1 2 0
x ∈IRn, u:IRn →IRn linear and |det(u)|=1 such that
0
|T(A )+A |1/n ≤C(|A |1/n+|A |1/n)
1 2 1 2
In particular, for the class of p-balls the constant C is universal (depending only on p).
We provethis theoreminsection 2. The key is to estimate certainentropynumbers. We will use
the convexity of quasi-normed spaces of Rademacher type r > 1, as well as interpolation results and
iteration procedures.
We want to thank Gilles Pisier for a useful conversationduring the preparationof this paper.
1. Notation and background
Throughoutthe paperX,Y,Z will denote finite dimensionalrealvectorspaces. Aquasi-normon
a real vector space X is a map k·k:X →IR+ such that
i) kxk>0 ∀ x6=0.
ii) ktxk=|t| kxk ∀ t∈IR,x∈X.
iii) ∃C ≥1 such that kx+yk≤C(kxk+kyk) ∀ x,y ∈X
If iii) is substituted by
iii’) kx+ykp ≤kxkp+kykp for x,y ∈X and some 0<p≤1,
k·k is called a p-norm on X. Denote by B the unit ball of a quasi-normed or a p-normed space.
X
The above observations concerning the p-convexification of our problem can be restated using
p-norm and quasi-norm notation. Recall that any compact balanced set with 0 in its interior is the
unit ball of a quasi-norm.
By the concavityofthe function tp, anyp-normisa quasi-normwithC =21/p−1. Conversely,by
theAoki-Rolewicztheorem,foranyquasi-normwithconstantC thereexistsp,namelyp=log−1(2C),
2
and a p-norm |·| such that |x|≤kxk≤41/p|x|, ∀ x∈X.
A set K ⊂ X is called p-convex if λx+µy, whenever x,y ∈ K, λ,µ ≥ 0, λp +µp = 1. Given
K ⊆ X, the p-convex hull (or p-convex envelope) of K is the intersection of all p-convex sets that
containK. Itisdenotedbyp-conv (K). Theclosedunitballofap-normedspace(X,k·k)willsimply
be called a p-ball. Any symmetric compact p-convex set in X with the origin as an interior point is
the p-ball associated to some p-norm.
We say that a quasi-normed space (X,k·k) is of (Rademacher) type q,0 < q ≤ 2 if for some
constant T (X)>0 we have
q
n n
1
k ε x k≤T (X)( kx kq)1/q, ∀x ∈X,1≤i≤n, ∀n∈IN
2n i i q i i
εiX=±1 Xi=1 Xi=1
Kalton, [Ka], proved that any quasi-normed space (X,k·k) of type q > 1 is convex. That is,
the quasi-norm k·k is equivalent to a norm and moreover,the equivalence constant depends only on
T (X), (for a more precise statement and proof of this fact see [K-S]).
q
Givenf,g:IN →IR+ wewritef ∼g ifthereexistsaconstantC >0suchthatC−1f(n)≤g(n)≤
Cf(n), ∀ n ∈ IN. Numerical constants will always be denoted by C (or C if it depends only on p)
p
although their value may change from line to line.
2
Letu:X →Y be a linearmapbetweentwoquasi-normedspacesandk ≥1. Recallthe definition
of the following numbers:
Kolmogorov numbers: d (u) = inf{kQ ◦ uk | S ⊂ Y subspace and dim(S) < k} where
k S
Q :Y →Y/S is the quotient map.
S
Covering numbers: For A ,A ⊂ X, N(A ,A ) = inf{N ∈ IN |∃ x ...x ∈ X such that A ⊂
1 2 1 2 1 N 1
(x +A )}.
1≤i≤N i 2
ESntropy numbers: e (u)=inf{ε>0|N(u(B ),εB )≤2k−1}
k X Y
The sequences {d (u)},{e (u)}are non-increasingandsatisfyd (u)=e (u)=kuk. If dim(X)=
k k 1 1
dim(Y)=nthend (u)=0forallk >n. Denotes eitherd ore . Foralllinearoperatorsu:X →Y,
k k k k
v:Y →Z we have s (v◦u)=kuks (v) and s (v◦u)=kvks (u), ∀k ∈IN (called the ideal property
k k k k
of s ) and
k
s (v◦u)≤s (v)s (u) ∀ k,n∈IN
k+n−1 k n
Thefollowingtwolemmascontainusefulinformationaboutthesenumbers. Thefirstoneextends
to the p-convex case its convex analogue due to Carl ([Ca]). Its proof mimics the ones of Theorem
5.1 and 5.2 in [Pi 1] (see also [T]) with minor changes. In particular we identify X as a quotient
of ℓ (I), for some I, and apply the metric lifting property of ℓ (I) in the class of p-normed spaces
p p
(see Proposition C.3.6 in [Pie]). The second one contains easy facts about N(A,B) and its proof is
similar to the one of Lemma 7.5. in [Pi 1] .
Lemma 1. For all α>0 and 0<p<1 there exists a constantC >0 such that for all linear map
α,p
u:X →Y, X,Y p-normed spaces and for all n∈IN we have
supkαe (u)≤C supkαd (u)
k α,p k
k≤n k≤n
Lemma 2.
i) For all A ,A ,A ⊂X, N(A ,A )≤N(A ,A )N(A ,A )
1 2 3 1 3 1 2 2 3
ii) For all t>0 and 0<p<1 there is C >0 such that for all X p-normed space of dimension n,
p,t
N(B ,tB )≤Cn .
X X p,t
iii) For any A ,A ,K ⊂IRn, |A +K|≤N(A ,A )|A +K|.
1 2 1 1 2 2
|B |
iv) Let B ,B be p-balls in IRn for some p and B ⊂B ; then 1 ∼N(B ,B ).
1 2 2 1 1 2
|B |
2
For any B ⊆IRn p-ball the polar set of B is defined as
B◦ :={x∈IRn |hx,yi≤1, ∀y ∈B}
where h·,·i denotes the standard scalar product on IRn. Given B,D p-balls in IRn we define the
following two numbers:
s(B):=(|B|·|B◦|)1/n
and
|B+D| |B◦+D◦| 1/n
M(B,D):= ·
|B∩D| |B◦∩D◦|
(cid:18) (cid:19)
Observe that for any linear isomorphism u:IRn →IRn we have s(u(B))=s(B) and
M(u(B),u(D))=M(B,D).
Recall that s(Bℓnp)∼n−1/p ∼s(Bℓn2)1/p,0<p≤1 ([Pi 1] pg. 11).
The following estimates on these numbers are known:
3
a) [Sa]. For every symmetric convex body B ⊂ IRn, s(B) ≤ s(Bℓn2) with equality only if B is an
ellipsoid. (Blaschke-Santalo´’sinequality).
b) [B-M]. There exists a numerical constant C > 0 such that for any n ∈ IN and any symmetric
convex body B ⊂IRn, s(B)≥Cs(Bℓn2).
c) [Mil 1]. There exists a numerical constant C > 0 such that for any n ∈ IN and any symmet-
ric convex body B ⊂ IRn, there is an ellipsoid (called Milman ellipsoid) D ⊂ IRn such that
M(B,D)≤C,(Milman ellipsoid theorem).
2. Entropy estimates and reverse Brunn-Minkowski inequality
Wefirstintroducesomeusefulnotation: LetB ,B ⊂IRnbetwop-ballsandu:IRn →IRnalinear
1 2
map. We denote u:B → B the operator between p-normed spaces u:(IRn,k·k ) → (IRn,k·k )
1 2 B1 B2
where k·k is the p-norm on IRn whose unit ball is B .
Bi i
Proof of Theorem 1:
Let A ,A be two bodies in IRn such that p(A ), p(A ) ≥ p. It’s clear from the definition
1 2 1 2
that there exist two p¯-balls, B , B , (for instance, p¯ = p/2) and two measure preserving affine
1 2
transformations T , T , verifying
1 2
|T−1T (A )+A |≤|B +B |
2 1 1 2 1 2
and
|B |1/n+|B |1/n ≤C |A |1/n+|A |1/n .
1 2 p 1 2
(cid:16) (cid:17)
So, we only have to prove the theorem for p-balls.
In the convex case a way to obtain the reverse Brunn-Minkowski inequality is to prove that, for
any symmetric convex body B, there exists an ellipsoid D verifying |B|=|D| and
|B+∆|1/n ≤C|D+∆|1/n (2)
for any, say compact, subset ∆⊆IRn (C is an universal constant independent of B and n).
Indeed, let B ,B be two balls in IRn. Suppose w.l.o.g. that D , the ellipsoids associated to B
1 2 i i
satisfy u2Di =αiBℓn2, where ui are linear mappings with |det ui|=1 and |Bi|1/n =αi|Bℓn2|1/n.Then
|u B +u B |1/n ≤C2|u D +u D |1/n
1 1 2 2 1 1 2 2
=C2(α1+α2)|Bℓn2|1/n =C2(|B1|1/n+|B2|1/n)
In view of the preceding comments and of straightforward computations deduced from Lemma
2, in order to obtain (2) for p-balls it is sufficent to associate an ellipsoid D to each p-ballB ⊂IRn in
suchawaythatthecorrespondingcoveringnumbersverifyN(B,D),N(D,B)≤Cn forsomeconstant
C depending only on p.
It is important to remark now the fact that, what we deduce from covering numbers estimate is
that the ellipsoid D associated to B actually verifies the stronger assertion
C−1|B+∆|1/n ≤|D+∆|1/n ≤C|B+∆|1/n
foranycompactset∆inIRn,withconstantdependingonlyonp. Furthermore,theroleoftheellipsoid
can be played by any fixed p-ball in a “spetial position”.
Denote by Bˆ the convex hull of B.
Bydefinitionofe ,ife (id:B →D)≤λthenN(B,2λD)≤2n−1andbyLemma2-ii),N(B,D)≤
n n
λn. (Of course, the same can be done with N(D,B)). Therefore our problem reduces to estimating
entropy numbers. What we are going to prove is really a stronger result than we need, in the line of
Theorem 7.13 of [Pi 2].
4
Lemma 3. Given α>1/p−1/2,there exists a constantC =C(α,p) suchthat, for anyn∈IN and
for any p-ball B ∈IRn we can find an ellipsoid D ∈IRn such that
n α
d (D →B)+e (B →D)≤C
k k
k
(cid:16) (cid:17)
for every 1≤k≤n.
Proof of the Lemma. From Theorem 7.13 of [Pi 2] we can easily deduce the following fact: There
exists a constant C(α) > 0 such that for any 1 ≤ k ≤ n, n ∈ IN and any ball Bˆ ⊂ IRn, there is
ellipsoid D ⊂IRn such that the identity operator id:IRn →IRn verifies
0
n α n α
d (id:D →Bˆ)≤C(α) and e (id:Bˆ →D )≤C(α) (3)
k 0 k 0
k k
(cid:16) (cid:17) (cid:16) (cid:17)
For simplicity, since we are always going to deal with the identity operator, we will denote
id:B →B by B →B .
1 2 1 2
Let D be the ellipsoid associated to Bˆ in (3). It is well known [Pe], [G-K] that B ⊆ Bˆ ⊆
0
n1/p−1B. This means kB →Bˆk≤ 1 and kBˆ →Bk ≤ n1/p−1. Now, (3) and the ideal property of d
k
and e imply
k
n α n α
d (D →B)≤C(α)n1/p−1 and e (B →D )≤C(α) ∀ k ≤n
k 0 k 0
k k
(cid:16) (cid:17) (cid:16) (cid:17)
This let us to introduce the constant C as the infimum of the constants C > 0 for which the
n
conclusion of lemma 3 is true for all p-ball in IRn. Trivially C ≤ C(α) 1+n1/p−1 . Let D be an
n 1
almost optimal ellipsoid such that
(cid:0) (cid:1)
n α
d (D →B)≤2C
k 1 n
k
(4)
(cid:16)n(cid:17)α
e (B →D )≤2C
k 1 n
k
(cid:16) (cid:17)
for every 1≤k≤n.
Use the real interpolation method with parameters θ,2 to interpolate the couple id:B →B and
id:D →B. It is straightforwardfrom its definition that for B :=(B,D ) , we have
1 θ 1 θ,2
d (B →B)≤kB →Bk1−θ(d (D →B))θ ∀ k≤n
k θ k 1
and therefore,
n α θ
d (B →B)≤ 2C ∀ k ≤n
k θ n
k
(cid:16) (cid:16) (cid:17) (cid:17)
n α
Write λ = 4C . By definition of the entropy numbers, there exist x ∈ IRn such that
n i
k
2k−1 (cid:16) (cid:17)
B ⊂ x +2λD . But by perturbing λ with an absolute constant we can suppose w.l.o.g. that
i 1
i=1
[
x ∈ B. For all z ∈ B, there exists x ∈ B such that kz − x k ≤ 2λ. Also by p-convexity,
i i i D0
kz−x k ≤21/p.
i B
A general result (see [B-L] Ch. 3.) assures the existence of a constant C >0 such that
p
kxk ≤C kxk1−θkxkθ .
Bθ p B D1
Therefore, for all z ∈B, there exists x ∈B such that kz−x k ≤C λθ which means
i i Bθ p
n α θ
e (B →B )≤C 2C .
k θ p n
k
(cid:16) (cid:16) (cid:17) (cid:17)
5
2(1−p)
Since α > 1/p−1/2, then we can pick θ ∈ (0,1) such that < θ < min{1,1−1/2α}.
2−p
Then B has Rademacher type strictly bigger than 1 because 1−θ + θ <1.
θ p 2
By Kalton’sresultquotedbefore, we cansuppose thatB is a ballandtherefore we canapply to
θ
it (3) for γ =α(1−θ)>1/2 and assure the existence of another ellipsoid D such that
2
n γ n γ
d (D →B )≤C(γ) and e (B →D )≤C(γ) and ∀ k ≤n
k 2 θ k θ 2
k k
(cid:16) (cid:17) (cid:16) (cid:17)
Recall that d (D → B) ≤ d (D → B )d (B → B) and the same for the e ’s. Thanks to
2k−1 2 k 2 θ k θ k
the monotonicity of the numbers s we can use the what is known about s for all s . Using the
k 2k−1 k
estimates obtained above we get ∀ k ≤n,
n γ+αθ n γ+αθ
d (D →B)≤C(p,α)2θCθ and e (B →D )≤C(p,α)2θCθ .
k 2 n k k 2 n k
(cid:16) (cid:17) (cid:16) (cid:17)
Hence by the election of γ and by minimality we obtain C1−θ ≤C(p,α)2θ, and the conclusion of the
n
lemma holds.
///
The theorem follows now from the estimate we achieved in Lemma 3 and by Lemma 1. Indeed,
given any α>1/p−1/2, if D is the ellipsoid associated to B by Lemma 3, we have
nα
nαe (D →B)≤ supkαe (D →B)≤C(α,p)supkαd (D →B)≤C(α,p)supkα
n k k kα
k≤n k≤n k≤n
=C(α,p)nα
and so, e (D →B)≤C(α,p). On the other hand just take k =n in Lemma 3 and so, e (B →D)≤
n n
C(α,p).
Finally observe that since the constant C(α,p) depends only on p and α and we can take any
α>1/p−1/2 the thesis of the theorem as stated inmediately follows.
///
6
3. Concluding remarks
We conclude this note by stating the corresponding versions of a) Blaschke-Santalo´, b) reverse
Blaschke-Santalo´ and c) Milman ellipsoid theorem, cited in section 1, in the context of p-normed
spaces.
Proposition 1. Let 0<p ≤1. There exists a numerical constant C >0 such that for every p-ball
p
B ⊆IRn,
1/p
Cp s(Bℓn2) ≤s(B)≤s(Bℓn2)
and in the second inequality, equality(cid:0)holds if(cid:1)only if B is an ellipsoid
Proof: Denote by Bˆ the convex envelope of B. Since Bˆ◦ = B◦ we have s(B) ≤ s(Bˆ) ≤ s(Bℓn2). If
s(B)=s(Bℓn2), then Bˆ is an ellipsoid. We will show that B =Bˆ. Every x in the boundary of Bˆ can
be written as x = λ x ,x ∈ B, λ = 1; but since Bˆ is an ellipsoid, x is an extreme point of Bˆ
i i i i
and so x=x for some i that is x∈B. This shows B =Bˆ and we are done.
i
P P
1/n
|Bˆ|
For the first inequality, B ⊆Bˆ ⊆n1/p−1B easily implies ≤n1/p−1 and so,
|B|
!
1/n
|B| s(Bˆ)
s(B)=(|B|·|B◦|)1/n =(|B|·|Bˆ◦|)1/n = (|Bˆ|·|B◦|)1/n ≥
|Bˆ|! n1/p−1
≥ Cns1(/Bp−ℓn21) =Cn−1/p =Cp s(Bℓn2) 1/p
(cid:0) (cid:1)
///
1/p
The left inequality above is sharp since s(Bℓnp) = Cp s(Bℓn2) . The right inequality is also
sharp since every ball is a p-ball for every 0 < p < 1. And it is sharp even if we restrict ourselves to
(cid:0) (cid:1)
the class of p-balls which are not q-convex for any q > p, as it is showed by the following example:
Let ε > 0 and C be a relatively open cap in Sn−1 centered in x = (0,...,0,1) of radius ε. Write
ε
K =Sn−1\{C ∪−C }. The p-ball p-conv (K) is not q-convex for any q >p and we can pick ε such
ε ε
s(p-conv (K))
that ∼1.
s(Bℓn2)
Observe that the left inequality is actually equivalent to the existence of a constant C >0 such
p
1/n
|Bˆ|
that for every p-ball B, ≤ C n1/p−1 and by Lemma 2 iv), this is also equivalent to the
p
|B|
!
inequality N(Bˆ,B)≤C n1/p−1.
p
With respect to the Milman ellipsoid theorem we obtain
Proposition 2. Let 0<p <1. There exists a numerical constant C >0 such that for every p-ball
p
B there is an ellipsoid D such that M(B,D)≤C n1/p−1.
p
Proof: Given a p-ball B let D be the Milman ellipsoid of Bˆ. Then,
|B+D| |B◦+D| 1/n
M(B,D)= ·
|B∩D| |B◦∩D|
(cid:18) (cid:19)
1/n 1/n 1/n
|Bˆ+D| |Bˆ◦+D| |B+D| |Bˆ∩D|
= ·
|Bˆ∩D| |Bˆ◦∩D|! |Bˆ+D|! |B∩D|!
1/n
|Bˆ∩D|
≤M(Bˆ,D) ≤C n1/p−1
p
|B∩D|
!
7
///
The bound for M(B,D) is sharp. Indeed, if there was a function f(n) << n1/p−1 such that for
every a p-ball B there was an ellipsoid D with M(B,D)≤f(n), then
s(Bℓn2) = s(D) ≤(|B∩D|·|B◦∩D|)1/n ≤s(B)
f(n) f(n)
and we would have, s(B)≥ s(Bℓn2) >>n−1/p which is not possible.
f(n)
Acknowledgments. The authors are indebted to the referee for some fruitful comments which
led them to improve the presentation of the paper.
References
[B-L] Bergh, J. Lo¨fstr¨om, J.: Interpolation Spaces. An Introduction. Springer-Verlag, 223. Berlin
Heildelberg (1976).
[B-M] Bourgain,J., Milman, V.D.: On Mahler’s conjecture on the volume of a convex symmetric body
and its polar. Preprint I.H.E.S., March 1985.
[Br] Brunn, H.: U¨ber Ovale und Eifla¨chen. Innaugural dissertation. Mu¨nchen 1887.
[Ca] Carl B.: Entropy numbers, s-numbers, and eigenvalue problems. J. Funct. Anal. 41, 290-306
(1981).
[G-K] Gordon, Y., Kalton, N.J.: Local structure for quasi-normed spaces. Preprint.
[Ja] Jarchow,H.: Locally convex spaces. B.G. Teubner Stuttgart, (1981).
[Ka] Kalton,N.: Convexity,TypeandtheThreeSpaceProblem. StudiaMath. 69,247-287(1980-81).
[K-S] Kalton, N., Sik-Chung Tam: Factorization theorems for quasi-normed spaces. Houston J. Math.
19 (1993), 301-317.
[K-P-R] Kalton, N., Peck, N.T., Roberts, J.W. : An F-space sampler. London Math. Soc. Lecture Notes
89. Cambridge Univ. Press. Cambridge (1985).
[Lu] Lusternik, L.A.: Die Brunn-Minkowskische Ungleichung fu¨r beliebige messbare Mengen. C.R.
Acad.Sci.URSS 8, 55-58 (1935).
[Mil 1] Milman, V.D.: In´egalit´e de Brunn-Minkowsky inverse et applications ´a la th´eorie locale des
espaces norm´es. C.R. Acad.Sci.Paris 302,S´er 1, 25-28 (1986).
[Mil 2] Milman,V.D.: Isomorphicsymmetrizationsandgeometricinequalities. In“GeometricAspectsof
Functional Analysis” IsraelSem. GAFA. Lecture Notes in Mathematics, Springer,1317, (1988),
107-131.
[Min] Minkowski, H.:Geometrie der Zahlen. Teubner, Leipzig (1910).
[Pe] Peck,T.: Banach-Mazurdistancesandprojectionsonp-convexspaces. Math. Zeits. 177,132-141
(1981).
[Pie] Pietsch, A.: Operator ideals. North-Holland, Berlin 1979.
[Pi 1] Pisier, G.: The volume of convex bodies and Banach Space Geometry. Cambridge University
Press, Cambridge (1989).
[Pi 2] Pisier,G.: A new approachto severalresults of V. Milman. J. reine angew. Math. 393, 115-131
(1989).
8
[Ro] Rolewicz, S.: Metric linear spaces. MM 56. PWN, Warsaw (1972).
[Sa] Santalo´, L.A.: Un invariante af´ın para los cuerpos convexos de n dimensiones. Portugal Math.
8, 155-161(1949).
[T] Triebel,H.: Relationsbetweenapproximationnumbersandentropynumbers. JournalofApprox.
Theory 78, (1994), 112-116.
9