Minimal-Volume Shadows of Cubes

Journal of Functional Analysis 176, 317�330 (2000)

Minimal-Volume Shadows of Cubes1

M. I. Ostrovskii

Department of Mathematics, The Catholic University of America,Washington, District of Columbia 20064;

and Mathematical Division, Institute for Low Temperature Physics, 47 Lenin Avenue,310164 Kharkov, Ukraine

Communicated by G. Pisier

Received July 23, 1999; accepted June 2, 2000

We study the shape of minimal-volume shadows of a cube in a given subspace.First we prove an essentially known result that for every subspace L the set of mini-mal-volume shadows in L contains a parallelepiped. Our main result is that forsome subspaces there exist minimal-volume shadows that are far fromparallelepipeds with respect to the Banach�Mazur distance. � 2000 Academic Press

Key Words: cube; projection; minimal volume; the Banach�Mazur distance;compound matrix; totally unimodular matrix; Sobolev inequality; directed graph;Grothendieck theorem; absolutely summing operator.

Let Km/Rm be defined by Km=[(x1 , ..., xm) : |xi |�1 for everyi # [1, ..., m]]. We refer to Km as an m-cube. Let L be a linear subspace inRm and P: Rm � L be a linear projection onto L. The set P(Km) will becalled a shadow of Km in L. Using a compactness argument it can beproved that for every m # N and for every subspace L/Rm there exists alinear projection that minimizes the volume of P(Km). In such a case theset P(Km) will be called a minimal-volume shadow of K m in L.

It may happen that Km has many different minimal-volume shadows inL. The purpose of this paper is to study the shape of minimal-volumeshadows of cubes. It is known that among minimal-volume shadows in anarbitrary subspace there is always a parallelepiped (see Theorem 1). Ourmain result is that there exist minimal-volume shadows that are far fromparallelepipeds with respect to the Banach�Mazur distance. Such shadowscan be found by a simple and explicit construction; see the begining of theproof of Theorem 2.

doi:10.1006�jfan.2000.3641, available online at http:��www.idealibrary.com on

3170022-1236�00 �35.00

Copyright � 2000 by Academic PressAll rights of reproduction in any form reserved.

1 This research was started when the author was supported by an INTAS grant and by agrant of TU� BITAK. The work was finished while the author was visiting the University ofCalifornia at Riverside. The author thanks the Department of Mathematics of the Universityof California at Riverside and Professor B. L. Chalmers for their hospitality.

Initially this study was motivated by the study of sufficient enlargements(see [9]). Here we do not discuss this connection, because it is also anatural geometric problem.

The following result is essentially known. It is implicitly contained in [5,pp. 95�97]. We prove it because our proof is more direct than the proof in[5] and we use our proof in further considerations.

Theorem 1. Let L be a linear subspace in Rm. Let M be the set allminimal-volume shadows of Km in L. The set M contains a parallelepiped.

Proof. Denote by [ei]mi=1 the unit vector basis in Rm. Let n=dim L

and let

E=lin[ei(1) , ..., ei(m&n)],

where [i(1), ..., i(m&n)] is a subset of [1, ..., n], be such that L & E=[0].Let P be the projection of Rm onto L with kernel E. Then P(Km) is aparallelepiped. We endow Rm with the standard inner product and com-pute all volumes with the corresponding normalization. Let z1 , ..., zm # Rm

be such that zj=�mi=1 zi, j ei . By det[z1 , ..., zm] we mean the determinant of

the matrix [zi, j]mi, j=1 .

Let [x1 , ..., xn] be some orthonormal basis in L. Then

vol P(Km)=2n

|det[x1 , ..., xn , ei(1) , ..., ei(m&n)]|.

Suppose that E is chosen in such a way that

|det[x1 , ..., xn , ei(1) , ..., ei(m&n)]|

takes the maximal possible value.Let Q: Rm � L be another projection. Let q1 , ..., qm&n be an orthonormal

basis in its kernel. We have

vol Q(Km)=2n

|det[x1 , ..., xn , q1 , ..., qm&n]|

_ :[ j(1), ..., j(n)]/[1, ..., m]

|det[q1 , ..., qm&n , ej(1) , ..., ej(n)]|,

where the sum is over all n-element subsets of [1, ..., m]. (To prove this for-mula we first project the cube onto the orthogonal complement of the ker-nel of Q and use the well-known formula for the volume of a zonotope, see[13, Eq. (57)]. Then we use the previous formula.)

318 M. I. OSTROVSKII

In order to prove the theorem it is enough to show that

vol P(Km)�vol Q(Km). (1)

Inequality (1) is equivalent to the following inequality:

|det[x1 , ..., xn , q1 , ..., qm&n]|

�|det[x1 , ..., xn , ei(1) , ..., ei(m&n)]|

_ :[ j(1), ..., j(n)]/[1, ..., m]

|det[q1 , ..., qm&n , ej(1) , ..., ej(n)]|. (2)

By the Laplacian expansion (see [1, p. 78]) the determinant

det[x1 , ..., xn , q1 , ..., qm&n]

can be represented as

:I/[1, ..., m], *I=n

%I det XI det QI ,

where XI is the n_n-submatrix of [x1 , ..., xn] corresponding to I,QI is the corresponding (complementary) (m&n)_(m&n)-submatrix of[q1 , ..., qm&n], and [%I] are some signs.

It is easy to see that by the choice of [i(1), ..., i(m&n)] we have

|det[x1 , ..., xn , ei(1) , ..., ei(m&n)]|=maxI

|det XI |.

It is easy to see also that

:[ j(1), ..., j(n)]/[1, ..., m]

|det[q1 , ..., qm&n , ej(1) , ..., ej(n)]|=:I

|det QI |.

The inequality (2) follows. K

Our next purpose is to show that there exist minimal-volume shadowsthat are far from parallelepipeds.

Observe that each shadow is convex, closed, bounded and symmetricwith respect to 0. A shadow of Km in L has a non-empty interior in L.Hence it is a unit ball of some norm on L. With some abuse of terminologywe define the Banach-Mazur distance between a shadow and aparallelepiped as the Banach�Mazur distance between the normed spacecorrespondent to the shadow and ld

�, where d is the dimension of theshadow. We refer to [14] for basic facts on the Banach�Mazur distance.

319MINIMAL VOLUME SHADOWS

Convention. We use the term ball for a symmetric-with-respect-to-0,bounded, closed, convex body with nonempty interior in a finite dimen-sional linear space.

We say that two balls are affinely equivalent if there exists a linearoperator between the corresponding spaces that is a bijection of the balls.

A Minkowski sum of (finitely many) line segments in Rn is called azonotope (see [11] for basic facts on zonotopes). We shall considerzonotopes that are sums of line segments of the form [&x, x]. Suchzonotopes are balls according to our convention. Let a1 , ..., am be somecollection of vectors in Rn. The Minkowski sum

:m

i=1

[&ai , ai]

will be called the zonotope spanned by a1 , ..., am .

Theorem 2. Let 1<C<�. If ln k�2?3C2+3, k # N and n=k2, thenthere exists an (n&1)-dimensional subspace L of R2n such that the shadowP(K2n), where P is the orthogonal projection onto L, is a minimal-volumeshadow of K2n in L; and its Banach�Mazur distance to an (n&1)-dimensionalparallelepiped is �C.

Construction. Subspaces L satisfying the condition of the theorem canbe found in the following way. Let Gn be a two-dimensional discrete toruswith n vertices. (It means that Gn=Zk_Zk , where Zk is the group ofresidue classes of integers modulo k; vertices (x1 , y1) and (x2 , y2) are adja-cent if and only if either x1=x2 and y1= y2\1 (in Zk) or y1= y2 andx1=x2\1 (in Zk). We can visualize this graph drawing 2k circles on ausual torus; k of the circles are meridians and k are parallels.)

We consider Gn as a directed graph, edges are directed in an arbitraryway.

Let Mn be the incidence matrix of Gn , that is, an n_(2n) matrix whoserows and columns are indexed by the vertices and edges of Gn , respectively,and the column corresponding to an edge e has exactly two non-zeroentries: &1 in the row corresponding to the starting vertex of e and 1 inthe row corresponding to the end vertex of e.

We consider rows of Mn as vectors in R2n. Let L be the subspace of R2n

spanned by the rows of Mn . We claim that L has the property stated inTheorem 2.

We need the following notion:

Definition 3. A matrix A with real entries is called totally unimodularif determinants of all submatrices of A are equal to &1, 0 or 1.


Totally unimodular matrices is a very important object in integerprogramming. There exists a vast literature devoted to them (see Chapter19 in [12]). We need only the following observation that goes back toH.Poincare� : an incidence matrix of any directed graph is totallyunimodular. (See [12, pp. 274 and 378] for historical notes and a veryshort proof.) So, Mn is totally unimodular.

Lemma 4. Let A be a totally unimodular r_m matrix of rank l. Let Lbe the subspace in Rm spanned by rows of A. Let PL be the orthogonalprojection onto L. Then

(i) PL(K m) is a minimal-volume shadow of K m in L.

(ii) PL(K m) is affinely equivalent to the zonotope in Rr spanned bycolumns of A.

Proof. We rearrange the rows of A in order to get a matrix whose firstl rows are linearly independent. It is clear that the zonotope spanned by thecolumns of the obtained matrix is affinely equivalent to the zonotopespanned by the columns of A. Hence without loss of generality we mayassume that the first l rows of A are linearly independent.

By AT we denote the transpose of A. Let C be an upper-triangular r_rmatrix such that the first l columns of the product ATC form an ortho-normal basis in L and the remaining columns contain zeros only. The exist-ence of such matrices can be shown using the Gram�Schmidt orthonor-malization process. We denote by D the l_l submatrix of C correspondentto the first l rows and the first l columns. It is easy to see that D isinvertible.

Straightforward verification shows that the product ATCC TA is thematrix of PL with respect to the unit vector basis of Rm.

Let [x1 , ..., xm] be an orthonormal basis in Rm satisfying the followingcondition: vectors [x1 , ..., x l] are the first l columns of ATC. Writing[xk , ..., xs] we mean the matrix with columns xk , ..., xs .

We use results on compound matrices. We refer to [1, Chapter V] fornecessary definitions and results.

Let u=[ui] be an ( ml )-dimensional vector, where ui are l_l minors of

[x1 , ..., xl]. Since a compound matrix of an orthogonal matrix isorthogonal (see [1, Example 4 on p. 94]), then u is normalized (withrespect to the Euclidean norm). For the same reason the vector v=[vi] inthe ( m

m&l)-dimensional space, where vi are (m&l )_(m&l ) minors of[xl+1 , ..., xm], is also normalized.


Since the matrix [x1 , ..., xm] is orthogonal, its determinant is equal to\1. On the other hand, by the Laplacian expansion (see [1, p. 78]) thedeterminant is equal to

:

(ml )

i=1

%i u i vi

for proper signs %i and for proper ordering of ui and vi . (Observe that( m

l )=( mm&l).) Since u and v are normalized, it implies that either u i=% i vi

for every i or ui=&%i vi for every i.Now we let n=l, Q=PL , and [q1 , ..., qm&l]=[x l+1 , ..., xm] in the

argument of Theorem 1. We get: vol P(Km)=vol PL(Km) is equivalent to

|det[x1 , ..., xm]|=|det[x1 , ..., xl , e i(1) , ..., ei(m&l )]|

_ :[ j(1), ..., j(l )]/[1, ..., m]

|det[xl+1 , ..., xm , ej(1) , ..., ej(l)]|,

where [i(1), ..., i(m&l )] are chosen to maximize

|det[x1 , ..., x l , ei(1) , ..., ei(m&l )]|.

In terms of ui and vi this equality is

1=maxi

|ui| :

(ml )

i=1

|vi |. (3)

Let E be the matrix consisting of the first l rows of A. It is clear that Eis totally unimodular. It is easy to see that [x1 , ..., xl]=ETD. Therefore ui

is equal to det D, 0 or &det D for every i.

To prove equality (3) we observe that maxi |ui |=|det D|. Assume thatui=%i vi for every i (the case ui=&%i vi is similar). Then 1=|�i %i u i vi |=|det D| 2 } |, where | is the number of non-zero ui 's (=the number of non-zero vi 's). On the other hand,

|det D|2 } |=maxi

|ui| :

(ml )

i=1

|vi |.

It proves that PL(K m) is a minimal-volume shadow.The statement (ii) can be proved in the following way. Consider A as an

operator from Rm to Rr. The image of K m under A coincides with thezonotope spanned by the columns of A in Rr. This zonotope spans a


subspace of dimension l in Rr (because l=rank A). The operator PL=ATCC TA: Rm � Rm also has l-dimensional image. Therefore the restrictionof ATCCT to the range of A is an isomorphic embedding. Therefore theimage of Km under PL is affinely equivalent to the zonotope. K

Remark. The argument of Lemma 4 can be used to prove the followingstatement. Let F=BAT, where B is an m_m diagonal matrix with non-negative diagonal such that F has rank l. Let [q1 , ..., qm&l] be anorthonormal basis in the orthogonal complement of the linear span ofcolumns of F. Then the projection Q: Rm � L with the kernellin[q1 , ..., qm&l] satisfies

vol Q(K m)=vol P(Km)

and Q(Km) is affinely equivalent to the zonotope in Rr generated by therows of F.

Lemma 5. If ln k�2?3C2+3, then the Banach�Mazur distance betweenthe zonotope spanned by the columns of Mn and the parallelepiped of thesame dimension is �C.

Proof. Observe that the linear space spanned by the columns of Mn inRn is (n&1)-dimensional and it consists of all vectors whose sum of thecoordinates is equal to 0.

Let Xn be this space normed by the gauge functional of the zonotope.Observe that vertices of the zonotope spanned by a1 , ..., an are contained

in the set [�ni=1 % i ai : %i=\1] and that this set is contained in the

zonotope. Therefore the maximal value of a functional f over the zonotopeis equal to �n

i=1 | f (ai)|. Using this observation we can identify the dualspace Xn* of Xn with the space of functions on the set of vertices of Gn withzero average and with the norm

& f &*

= :utv

| f (u)& f (v)|,

where utv means that u and v are adjacent in Gn .We need to estimate the Banach�Mazur d(Xn , ln&1

� ) from below. Sinced(X, Y)=d(X*, Y*) for every finite-dimensional spaces X and Y, then

d(Xn , ln&1� )=d(Xn* , l n&1

1 ).

To estimate the distance d(Xn*, l n&11 ) we use the approach that goes

back to J. Lindenstrauss and A. Pe*czyn� ski (see [8, Proposition 6.1]).


Recall that the 2&summing norm of an operator T: X � Y is defined tobe the smallest constant C satisfying the condition

\ :n

i=1

&Txi &2+1�2

�C sup{\ :n

i=1

(!(xi))2+1�2

: ! # X*, &!&�1=for every collection [x1 , ..., xn] # X. The 2-summing norm of T is denotedby ?2(T ).

Let T: Z � H be a non-zero operator, where H is a Hilbert space anddim Z=n. The dual form of the ``little Grothendieck theorem'' (see [10,p. 57]; this form of the Grothendieck theorem [6] was proved in [4])implies that

d(Z, l n1)�\2

?+1�2 ?2(T )

&T&. (4)

So we need to find a Hilbert space H and an operator T: Xn* � H with``large'' ratio ?2(T)�&T&.

With this purpose in mind we introduce the norm

& f &2=\:v

( f (v))2+1�2

on the space of all functions on the set of vertices of Gn . We denote theobtained normed space by l2(Gn).

Let In be the identical embedding of Xn* into l2(Gn).To estimate the norm of this embedding from above we use a Sobolev

type inequality due to F. R. K. Chung and S.-T. Yau [3, Theorem 1](see, also, [2, p. 165]).

We need the following definitions.

Definition 6. Let G be a graph. By dv we denote the degree of a vertexv. Let X be some set of vertices of a graph G. Let vol X :=�v # X dv . Thenumber of edges joining X and its complement X� is denoted by |E(X, X� )|.We say that G has isoperimetric dimension $ with isoperimetric constantc$ if

|E(X, X� )|�c$(vol X ) ($&1)�$

whenever vol X�vol X� . The constant c$ depends on $ only.

Definition 7. A graph G is called k-regular if dv=k for every v.

We need the following special case of [3, Theorem 1].


Theorem 8. Let G be a connected k-regular graph with isoperimetricdimension 2 and isoperimetric constant c2 . Let f be a function on the set ofvertices of G with zero average. Then

:utv

| f (u)& f (v)|�c2

k1�2

2 \:v

( f (v))2+1�2

.

Observe that Gn is 4-regular. To apply Theorem 8 to Gn we need toestimate the isoperimetric constant c2 for this graph. Since the author hasnot found a proper reference, we present such an estimate (with the bestpossible constant).

Sublemma. The graph Gn has isoperimetric dimension 2 with constant- 2.

Proof. Let X be a set of vertices of Gn with *X�k2�2. Sets of verticesof the form

[(x, 0), (x, 1), (x, 2), ..., (x, k&1)]

will be called meridians and sets of the form

[(0, y), (1, y), (2, y), ..., (k&1, y)]

will be called parallels.Let m1 be the number of meridians contained in X and let m2 be the

number of meridians intersecting X. Let p1 be the number of parallelscontained in X and let p2 be the number of parallels intersecting X.

It is easy to see that

|E(X, X� )|�2(m2&m1)+2( p2& p1).

We have also *X�m1k and *X�p1 k. Hence m1�k�2 and p1�k�2.We have three possibilities:

(a) Both m1 and p1 are nonzero.

(b) Exactly one of the numbers m1 and p1 is nonzero.

(c) m1= p1=0.

(a) In this case m2= p2=k. Hence

|E(X, X� )|�2(m2&m1)+2( p2& p1)�2 \k&k2++2 \k&

k2+

=2k�2 - 2 \k2

2 +1�2

�2 - 2 (*X )1�2=- 2 (vol X )1�2.


(b) We consider the case m1 {0 and p1=0 (the case p1 {0 andm1=0 is similar). In this case p2=k and

|E(X, X� )|�2p2=2k�- 2 (vol X)1�2.

(c) In this case *X�m2p2 and

|E(X, X� )|�2(m2+ p2)�4 - m2p2 �4(*X )1�2=2(vol X )1�2. K

Remark. If k is even and X is the union of k2 meridians (or k

2 parallels),then

|E(X, X� )|=2k=- 2 (vol X )1�2.

So the constant - 2 is the best possible.

By Theorem 8 we get & f &*

�- 2 22 & f &2 for every f with average 0.

Hence &In&� 1- 2

.To estimate ?2(In) from below we use the approach developed by

S. V. Kislyakov [7] for continuous case.Let p be the integer part of k&1

2 . We introduce a family [ fs, t] ps, t=1 of

functions on Gn in the following way. We consider Gn as [0, ..., k&1]_[0, ..., k&1] and let

fs, t (x, y)=1

k(s+t)sin \2?s

kx+ sin \2?t

ky+ .

Observe that

?2(In)�(�s, t & fs, t&2

2)1�2

sup[(�s, t |!( fs, t)|2)1�2 : ! # (Xn*)*, &!&�1].

So we need to estimate the quantity

sup {\:s, t

|!( fs, t)|2+1�2

: ! # (X n*)*, &!&�1= (5)

from above and the quantity

\:s, t

& fs, t&22+

1�2

(6)

from below.


To estimate (6) we observe that

& fs, t&2=1

k(s+t) \ :k&1

x, y=0

sin2 \2?sk

x+ sin2 \2?tk

y++1�2

=1

k(s+t) \ :k&1

x=0

sin2 \2?sk

x+ :k&1

y=0

sin2 \2?tk

y++1�2

(we use the fact that 1�s, t< k2)

=1

k(s+t) \k2

k2+

1�2

=1

2(s+t).

Hence

\ :p

s, t=1

& fs, t&22+

1�2

�12 \

122+

232+ } } } +

p( p+1)2+

1�2

�12

(ln k&3)1�2.

To estimate (5) we observe that

:utv

| f (u)& f (v)|�- 2 k \ :utv

( f (u)& f (v))2+1�2

.

The right-hand side in this inequality is a hilbertian norm on Xn* inducedby the inner product

( f, g)=2k2 \ :k&1

x, y=0

( f (x+1, y)& f (x, y))(g(x+1, y)& g(x, y))

+ :k&1

x, y=0

( f (x, y+1)& f (x, y))(g(x, y+1)& g(x, y))+ , (7)

where (k&1)+1=0.We denote by Hn the corresponding Hilbert space. We shall use (4) for

H=Hn . Since the natural embedding of Hn into Xn* has norm 1, then thesupremum in (5) is not greater than

sup {\:s, t

|!( fs, t)|2+1�2

: ! # (Hn)*, &!&�1= , (8)

where the norm is in (Hn)*.To estimate this supremum we show that the functions [ fs, t] p

s, t=1 areorthogonal with respect to the inner product (7).


We have

( fs, t , fs$, t$)=2k2 \ :k&1

x, y=0

( fs, t (x+1, y)& fs, t (x, y))

_( fs$, t$(x+1, y)& fs$, t$(x, y))

+ :k&1

x, y=0

( fs, t (x, y+1)& fs, t (x, y))

_( fs$, t$(x, y+1)& fs$, t$(x, y))+ .

We shall show that the first sum is equal to zero (the same argumentworks for the second sum also).

:k&1

x, y=0

( fs, t (x+1, y)& fs, t (x, y))( fs$, t$(x+1, y)& fs$, t$(x, y))

=1

k(s+t)1

k(s$+t$):

k&1

x, y=0

2 sin \2?sk

12+ cos \2?s

k \x+12++ sin \2?t

ky+

_2 sin \2?s$k

12+ cos \2?s$

k \x+12++ sin \2?t$

ky+

=4

k2(s+t)(s$+t$)sin \2?s

k12+ sin \2?s$

k12+

_ :k&1

x=0

cos \2?sk \x+

12++ cos \2?s$

k \x+12++

_ :k&1

y=0

sin \2?tk

y+ sin \2?t$k

y+ .

By use of the fact that 1�s, s$, t, t$< k2 it is easy to show that if s{s$, then

the first sum in the last product is equal to 0, and if t{t$, then the secondsum is equal to 0.

Since the functions [ fs, t] ps, t=1 are orthogonal with respect to the inner

product (7), then the supremum in (8) is not greater than

maxs, t

- ( fs, t , fs, t) .


Using the computation above we get

( fs, t , fs, t)=2k2 \ 1k(s+t)+

2

\ :k&1

x, y=0

4 sin2 \2?sk

12+

_cos2 \2?sk \x+

12++ sin2 \2?t

ky+

+ :k&1

x, y=0

4 sin2 \2?tk

12+ cos2 \2?t

k \ y+12++ sin2 \2?s

kx++

(we use the inequality |sin z|�|z| )

�8

(s+t)2 \\?sk +

2

:k&1

x=0

cos2 \2?sk \x+

12++ :

k&1

y=0

sin2 \2?tk

y++\?t

k +2

:k&1

x=0

sin2 \2?sk

x+ :k&1

y=0

cos2 \2?tk \ y+

12+++

(each sum is equal to k2 (since 1�s, t< k

2))

=8

(s+t)2 \\?sk +

2 k2

4+\?t

k +2 k2

4 +�2?2.

Hence the suprema in (8) and (5) are not greater than - 2?. Therefore

?2(In)�12 (ln k&3)1�2

- 2 ?and

d(Xn , l n&1� )=d(Xn*, l n&1

1 )�\2

?+1�2 ?2(In)

&In &�

(ln k&3)1�2

- 2 ?3�2�C. K

Proof of Theorem 2. Consider the subspace L in R2n spanned by therows of Mn . By Lemma 4 the image of K2n under the orthogonal projectiononto L is a minimal-volume shadow. By the same lemma this shadow isaffinely equivalent to the zonotope spanned by the columns of Mn . ByLemma 5 the Banach�Mazur distance between the zonotope and aparallelepiped is �C. K

REFERENCES

1. A. C. Aitken, ``Determinants and Matrices,'' reprint of the 4th ed., Greenwood Press,Westport, CT, 1983.

2. F. R. K. Chung, ``Spectral Graph Theory,'' Amer. Math. Soc., Providence, RI, 1997.


3. F. R. K. Chung and S.-T. Yau, Eigenvalues of graphs and Sobolev inequalities, Combin.Probab. Comput. 4 (1995), 11�26.

4. Y. Gordon and D. R. Lewis, Banach ideals on Hilbert space, Studia Math. 54 (1975),161�172.

5. Y. Gordon, M. Meyer, and A. Pajor, Ratios of volumes and factorization through l� ,Illinois J. Math. 40 (1996), 91�107.

6. A. Grothendieck, Re� sume� de la the� orie me� trique des produits tensoriels topologiques, Bol.Soc. Mat. Sa~ o-Paulo 8 (1956), 1�79.

7. S. V. Kislyakov, Sobolev imbedding operators and the nonisomorphism of certain Banachspaces, Funct. Anal. Appl. 9 (1975), 290�294.

8. J. Lindenstrauss and A. Pe*czyn� ski, Absolutely summing operators in Lp-spaces and theirapplications, Studia Math. 29 (1968), 275�326.

9. M. I. Ostrovskii, Projections in normed linear spaces and sufficient enlargements, Arch.Math. 71 (1998), 315�324.

10. G. Pisier, ``Factorization of Linear Operators and Geometry of Banach Spaces,'' Amer.Math. Soc., Providence, RI, 1986.

11. R. Schneider and W. Weil, Zonoids and related topics, in ``Convexity and Its Applications''(P. M. Gruber and J. M. Wills, Eds.), pp. 296�317, Birkha� user, Basel�Boston�Stuttgart,1983.

12. A. Schrijver, ``Theory of Linear and Integer Programming,'' Wiley, New York, 1986.13. G. C. Shephard, Combinatorial properties of associated zonotopes, Canadian J. Math. 26

(1974), 302�321.14. N. Tomczak-Jaegermann, ``Banach�Mazur Distances and Finite-Dimensional Operator

Ideals,'' Pitman Monographs and Surveys in Pure and Applied Mathematics, Vol. 38,Longman Scientific 6 Technical, New York, 1989.


Documents

Minimal-Volume Shadows of Cubes