geometry of the
The Banach–Mazur compactum provides the natural framework for a quantitative study of finite-dimensional normed spaces with particular emphasis on the role of the dimension. Since every -dimensional normed real vector space is easily seen to be isometric to equipped with a suitable norm , the typical normed space discussed here will be . The unit ball of is a symmetric convex body. Conversely, if is a symmetric convex body in , then induces on the norm and the space has as its unit ball. Thus, the study of finite-dimensional normed spaces is parallel to that of symmetric convex bodies. If one assumes that an inner product defines the Euclidean structure on , then the dual space of (cf. also Adjoint space) has its norm given by . The unit ball of is the polar body of .
Let and be two -dimensional normed spaces. It is well-known that and are isomorphic (cf. Isomorphism). The Banach–Mazur distance
measures how well-isomorphic the two spaces are. An equivalent geometric interpretation is that is the smallest possible for which there exists a linear invertible transformation of such that . It is easy to check that for every and : ( is symmetric); if and only if is isometric to ; and , where , are the dual spaces of , , respectively.
Consider the set of all equivalence classes of -dimensional normed spaces, where is equivalent to if and only if and are isometric. Then it is not hard to check that becomes a contractible compact metric space with metric : the triangle inequality is a consequence of
which can be easily verified for every . The metric space is usually called the -dimensional Banach–Mazur compactum (or Minkowski compactum). In the sequel, instead of , the "multiplicative" distance on is considered.
The structure of the Banach–Mazur compactum is still to be explored. There are many interesting quantitative questions one may ask and, besides a few fundamental results, most of these questions remain open. Below, some basic problems and final or partial answers to them are given. Several other important topics are not mentioned here: the interested reader is referred to the books [a39], [a43], [a44], [a58], and to the survey papers [a21], [a29], [a30], [a38], [a41], and [a53].
The diameter of the compactum.
The first classical result on the geometry of is John's theorem [a26] on the maximal possible distance to the Euclidean space . Starting with an -dimensional space , F. John considered the linear transformation of for which the Euclidean unit ball is the ellipsoid of minimal volume containing . Using a variational argument, he obtained the following precise information about :
ii) there exist contact points of and (i.e., ), and such that the following representation of the identity holds: for every ,
In particular, i) gives an upper bound for (John's theorem): for every .
This estimate is sharp: one can see that . John's theorem and a direct application of the multiplicative triangle inequality (a1) give some first information on the diameter of :
A most natural question arising from (a2) is to determine . The exact value of this quantity is known only when : , by results of E. Asplund [a1] and W. Stromquist [a48]. Even the question of determining up to constants the order of growth of as a function of remained open for a long time. The answer was given by E.D. Gluskin [a18] in 1981 (Gluskin's theorem): There exists an absolute constant such that for every .
Therefore, the diameter of the Banach–Mazur compactum is of the order of . In his solution of the problem, Gluskin does not describe the pair of spaces with explicitly (in fact, no explicit example of two spaces with distance considerably larger than is known (1996)).
The existence of a pair with distance of the order of has been established by probabilistic arguments. Gluskin [a21] describes his idea as follows: Consider the pair and let be a random matrix whose entries are independent standard Gaussian variables. It is not hard to see that the inequality holds with high probability for matrices of this form. This suggests that by "spoiling" the space a bit it is possible to obtain spaces and such that for every linear isomorphism of . This is actually the case.
Gluskin considered spaces whose unit ball is a symmetric convex body of the form , where is the standard orthonormal basis of and , , are independent random points chosen from the Euclidean unit sphere equipped with the rotationally invariant probability measure . If is a pair of independent "Gluskin spaces" , one can show that the probability
is exponentially small, uniformly in . Combining this with exact estimates on the cardinality of -nets in suitable spaces of operators one gets that, for most pairs , the inequality holds true on a sufficiently fine net in . A standard argument, depending, however, on the precise estimates above, allows one to pass from the net to an arbitrary , thus showing that with .
Besides settling the question on the diameter of , Gluskin's method of considering random spaces proved to be extremely influential and fruitful. Random spaces provided the right framework for establishing "pathological behaviour" and solving several other major open problems in the asymptotic theory of finite-dimensional normed spaces. An example of this is Szarek's finite-dimensional analogue of Enflo's example of a space without the approximation property [a12]: There exist -dimensional normed spaces whose basis constant is of the order of [a51]. See also [a19], [a32] and subsequent work of S.J. Szarek and P. Mankiewicz, in which Gluskin spaces play a central role.
Extremal problems for special families of spaces.
In many important cases, the Banach–Mazur distance is significantly smaller than . For example, when both and are -spaces () one has the following classical Gurarii–Kadec–Macaev estimates [a25]:
i) if or , then ;
ii) if , then , where are absolute constants and .
Much work has been done in the direction of estimating for some important families . One of the methods invented for dealing with such problems is the method of random orthogonal factorization, which has its origin in work of N. Tomczak-Jaegermann [a55] and Y. Benyamini and Y. Gordon [a3]. The idea is to use the average of with respect to the probability Haar measure on the orthogonal group as an upper bound for . Technically, this process depends upon passing from to matrices whose entries are independent standard Gaussian variables and then use the theory of Gaussian processes to estimate this average.
for every . This estimate may be viewed as a generalization of the estimates for given above. This was further improved by J. Bourgain and V.D. Milman [a5] to
Two more results obtained by the same method and having an obvious geometric flavour are as follows.
Suppose that and are two -dimensional spaces such that and have few extreme points, in the sense that their cardinality is bounded by , where is a fixed positive number. Then
where is a constant depending only on (see [a8]).
Let be an -dimensional space. It has been shown [a5] that the distance from to its dual is "small" . More precisely, for some constant ,
The best-possible exponent of is not yet known (1996).
Both these results indicate that the distance between two spaces whose unit balls are "quite different" as convex bodies, should not be of the order of the diameter of . Recall that in Gluskin's theorem the spaces and with distance were of the same nature ( "spoiled" -spaces).
Now, let to be the family of all -symmetric spaces. Such a space possesses a basis with the property that for every choice of scalars , every choice of signs , and every permutation of the set of indices ,
Gluskin [a20] and Tomczak-Jaegermann have proved that if , then is bounded by up to some factor logarithmic in . Finally, Tomczak-Jaegermann [a56] succeeded to remove this extra factor, thus answering completely the following question on the maximal distance between -symmetric spaces (Tomczak-Jaegermann theorem): For two -symmetric spaces one has , where is an absolute constant.
Note that every is a -symmetric space, therefore the estimate above is clearly optimal (consider the distance between and ).
Another important family of spaces in is the family of -unconditional spaces. These are spaces for which there exists a basis with the property that for every choice of scalars and every choice of signs ,
One naturally asks for the order of
The final answer to this problem is not known (1996): it is conjectured that the right order is close to . J. Lindenstrauss and A. Szankowski [a31] have shown that it has to be smaller than : There exists a constant such that whenever and are two -unconditional spaces in , then for every , where is a constant depending only on . The constant obtained in [a31] is given by a complicated expression. One knows that and numerical calculations show that it should be .
Radius of the compactum with respect to a fixed centre.
Let , and let be the radius of the Banach–Mazur compactum with respect to , defined by
In this terminology, John's theorem states that . It is natural to ask about sharp analogues of this result if is replaced by other standard -dimensional spaces. In this direction, A. Pelczynski [a41] asked a question of obvious geometric importance: What is the order of the maximal possible "Banach–Mazur distance to the cube" . Until recently, the only information available was that , a consequence of John's theorem. It turns out that none of these two estimates is sharp: ([a7]; the Bourgain–Szarek estimate). On the way to this estimate, very interesting information on the relation between a symmetric convex body and its minimal volume ellipsoid has been obtained. Assuming that is the minimal ellipsoid of , as in John's argument, it was proved that for every one can choose , , among the contact points of and , such that for every choice of scalars ,
The important part in this string of inequalities is, of course, the first one. This is a strengthened version of the classical Dvoretzky–Rogers lemma [a11], which implied a similar inequality only for . It can also be stated in the form of a "proportional factorization result" (proportional Dvoretzky–Rogers factorization, [a7]): Let be an -dimensional space. For every one can find an and two operators , , such that the identity is written as and , where is a function depending only on the proportion .
Using this result, Bourgain and Szarek gave a final answer to the problem of the uniqueness, up to constant, of the centre of the Banach–Mazur compactum. This can be made a precise question as follows: Does there exist a function , , such that for every with one must have ? In other words, are all the "asymptotic centres" of the Banach–Mazur compactum close to the Euclidean space? The answer is negative and the main tool in the proof is the proportional Dvoretzky–Rogers result (non-uniqueness of an asymptotic centre for ; [a7]): Let , where . Then for some absolute constant, but . Therefore, there exist asymptotic centres of the Banach–Mazur compactum with distance to of the order of .
The same inequality allowed Bourgain and Szarek to show that . It is now known [a54], [a17] that (a3) holds true with . This gives a better upper bound for , which, however, does not seem to give the right order of magnitude (Banach–Mazur distance to the cube; [a52], [a16]): There exist two absolute constants such that
The lower bound , due to Szarek, shows that and are not asymptotic centres of the Banach–Mazur compactum (these are actually essentially the only explicit examples of spaces for which this property has been established). In Szarek's work, the space with distance to of the order of is once again a Gluskin space. So, the problem of the distance to the cube remains open (even without a strong conjecture about what the order of should be, 1996). The question of the best possible exponent of in the proportional Dvoretzky–Rogers factorization is also open (1996): by [a17] and [a46] it must lie between and .
Sections and projections of a symmetric convex body: distance to Euclidean space.
Some fundamental results on the local structure (i.e., the structure of subspaces and quotients) of finite-dimensional normed spaces are closely related to the Banach–Mazur compactum. The starting point of the asymptotic theory of finite-dimensional normed spaces is Dvoretzky's theorem [a9], [a10] on almost spherical sections of symmetric convex bodies: For every and every -dimensional normed space , there exist an integer and a subspace of with , such that
In other words, every symmetric convex body in has sections which are almost ellipsoids and whose dimension can be chosen to increase to infinity as the dimension of the original body tends to infinity. Being probabilistic in nature, the proof actually gives that most of the -dimensional subspaces of are almost Euclidean. The dependence on the dimension (first obtained by Milman in [a34]) is exact, as one can see by considering . There are several other proofs of Dvoretzky's theorem: see [a13], [a49]. The dependence on (established in [a22], [a47]) does not seem to be the right one: see [a37] for related remarks and conjectures.
A rough description of Milman's argument, which was further exploited in [a15], runs as follows: consider the quantity
where is the rotationally invariant probability measure on the Euclidean unit sphere . Without loss of generality one may assume that is the ellipsoid of maximal volume contained in . Then, is a -Lipschitz function on and the isoperimetric inequality on the sphere implies that the values of are highly concentrated around its expectation (cf. also Isoperimetric inequality). One can extract subspaces of dimension such that up to for all . The second main tool is the Dvoretzky–Rogers lemma (in a form dual to (a3)). A simple computation based on the information given by (a3) shows that, under the above hypotheses, has to be at least .
In many cases one has an estimate for which allows almost Euclidean subspaces of of dimension even proportional to . See [a15], where this is verified for , . A different approach, which has its origin in [a27] and was further simplified and generalized in [a50], shows that this is a common property of all spaces whose unit ball has "small" volume ratio (theorem on volume ratio and Euclidean sections): Let be an -dimensional normed space and set
where the is taken over all ellipsoids contained in . For every there exist subspaces with , such that
Note that if is bounded by a constant , then this theorem gives subspaces of of any proportional dimension and isomorphic to the Euclidean space up to a constant depending only on and (independent of the dimension ). A simple computation shows that the volume ratio of all spaces , , is uniformly bounded.
A crucial inequality in the same direction of a "proportional theory" of finite-dimensional spaces is Milman's low -estimate, [a35], which holds for every space : There exists a function such that for every and every one can find a subspace of with and satisfying
for all .
Again, the proof is probabilistic in nature and shows that this estimate holds true for most subspaces of a given dimension. There are several proofs of the Milman inequality; see [a40] and [a23] for the best possible dependence on : . Note that is half the mean width of the unit ball of and that (a4) is equivalent to
Thus, the geometric interpretation of Milman's inequality is that the diameter of the proportional sections of a symmetric convex body is controlled by the mean width of the body up to a function depending only on the proportion .
An important consequence of Milman's inequality is Milman's quotient-of-subspace theorem, [a35], which states that, starting with any space and any , one can find a quotient of a subspace of with dimension and which is Euclidean up to a constant depending only on . This should be compared with the -dimension of the Euclidean subspaces in Dvoretzky's theorem and the fact that the corresponding dimension for Euclidean quotients does not exceed in general. More precisely (the Milman quotient-of-subspace theorem): Let and . There exist subspaces such that and
Crucial results from [a14], [a28] and [a42] always allow one to have logarithmic in for a suitable choice of the Euclidean structure in . Then, an iteration scheme based on double applications of Milman's inequality completes the proof.
The quotient-of-subspace theorem has found several applications even in the context of classical convexity, such as the inverse Santaló inequality and the inverse Brunn–Minkowski inequality.
It was proved by W. Blaschke for , and by L. Santaló for general , that for every symmetric convex body in , where is the polar body of . The quotient-of-subspace theorem enabled Bourgain and Milman [a6] to prove that
for every body , where , are positive absolute constants. This shows that the affinely invariant volume product is the same for all symmetric convex bodies, up to an absolute constant.
The classical Brunn–Minkowski inequality (cf. also Brunn–Minkowski theorem) states that for every pair of convex bodies in . It is easy to check that, in general, no inverse to this inequality can hold. However, Milman [a36] proved the following: There exists a mapping sending each symmetric convex body to some linear transformation such that for every pair of symmetric convex bodies and in the following inequality holds:
where is an absolute constant. The body is called an -position of .
An example of the interaction between the local structure of normed spaces and the geometry of the Banach–Mazur compactum is the isomorphic finite-dimensional version of the homogeneous space problem. A question raised by S. Banach in [a2] is whether an -dimensional normed space all -dimensional subspaces of which are pairwise isometric for some , must be isometric to the Euclidean space. In most cases this has been answered in the affirmative by M. Gromov [a24].
An isomorphic version of the same problem was studied by Bourgain [a4]. He showed that there are a and a function such that if is an -dimensional space with for every pair of -dimensional subspaces and of , then .
Mankiewicz and Tomczak–Jaegermann completed this result. They proved in [a33] that there is a function such that for every space all -dimensional subspaces of which are -isomorphic. The positive answer to the question involves most of the basic methods and results mentioned in the previous sections.
Weak Banach–Mazur distance.
Tomczak–Jaegermann [a57] has introduced a different way of measuring the distance between two -dimensional normed spaces and . She first considered the weak factorization norm of the identity
where the is taken over all measure spaces and all mappings , such that . Then, one symmetrizes to define the weak distance between and as
One can easily check that for all . On the other hand, one can see that, with high probability, the weak distance between two Gluskin spaces is bounded by . So, the weak distance may differ much from the Banach–Mazur distance. Actually, M. Rudelson [a45] has given an example of a pair of spaces with weak distance less than and Banach–Mazur distance of order .
The diameter of with respect to the weak distance is much smaller than . Rudelson [a45] has proved that
for every pair . It has been conjectured that the weak distance in is always bounded by .
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