Vitali Milman

λ1, Isoperimetric inequalities for graphs, and superconcentrators

Abstract A general method for obtaining asymptotic isoperimetric inequalities for families of graphs is developed. Some of these inequalities have been applied to functional analysis. This method uses the second smallest eigenvalue of a certain matrix associated with the graph and it is the discrete version of a method used before for Riemannian manifolds. Also some results are obtained on spectra of graphs that show how this eigenvalue is related to the structure of the graph. Combining these results with some known results on group representations many new examples are constructed explicitly of linear sized expanders and superconcentrators.

The Santaló point of a function, and a functional form of the Santaló inequality

Let L(f) denote the Legendre transform of a function f : ℝ n → ℝ. A theorem of K. Ball about even functions is generalized, and it is proved that, for any measurable function f ≥ 0, there exists a translation f(x) = f(x−a) such that

Extremal problems and isotropic positions of convex bodies

LetK be a convex body in ℝn and letWi(K),i=1, …,n−1 be its quermassintegrals. We study minimization problems of the form min{Wi(TK)|T ∈ SLn} and show that bodies which appear as solutions of such problems satisfy isotropic conditions or even admit an isotropic characterization for appropriate measures. This shows that several well known positions of convex bodies which play an important role in the local theory may be described in terms of classical convexity as isotropic ones. We provide new applications of this point of view for the minimal mean width position.

On type of metric spaces

Families of finite metric spaces are investigated. A notion of metric type is introduced and it is shown that for Banach spaces it is consistent with the standard notion of type. A theorem parallel to the Maurey-Pisier Theorem in Local Theory is proved. Embeddings of Zp-cubes into the ¡i-cube (Hamming cube) are discussed.

Better expanders and superconcentrators

Abstract Explicit construction of families of linear expanders and superconcentrators is relevant to theoretical computer science in several ways. Here we construct better expanders than those previously known and use them to construct explicitly n-superconcentrators with ≈ 122.74n edges; much less than the previous most economical construction.

Chapter 17 – Euclidean Structure in Finite Dimensional Normed Spaces

This chapter discusses the results that stand between geometry, convex geometry, and functional analysis. The chapter describes the family of n -dimensional normed spaces and discusses the study on the asymptotic behavior of their parameters, as the dimension n grows to infinity. This theory grew out of functional analysis. In fact, it may be viewed as the most recent one among many examples of directions in mathematics that were born inside this field during the twentieth century. The influence of the ideas of functional analysis on mathematical physics not only on differential equations but also on classical analysis was enormous. The great achievements and successful applications to other fields led to the creation of new directions (among them, algebraic analysis, noncommutative geometry, and the modem theory of partial differential equations) that, in a short time, became autonomous and independent fields of mathematics.

Eigenvalues, Expanders And Superconcentrators

Explicit construction of families of linear expanders and superconcentrators is relevant to theoretical computer science in several ways. There is essentially only one known explicit construction. Here we show a correspondence between the eigenvalues of the adjacency matrix of a graph and its expansion properties, and combine it with results on Group Representations to obtain many new examples of families of linear expanders. We also obtain better expanders than those previously known and use them to construct explicitly n-superconcentrators with 157.4 n edges, much less than the previous most economical construction.

Banach spaces with a weak cotype 2 property

We study the Banach spacesX with the following property: there is a numberδ in ]0,1[ such that for some constantC, any finite dimensional subspaceE ⊂X contains a subspaceF ⊂E with dimF≧δ dimE which isC-isomorphic to a Euclidean space. We show that if this holds for someδ in ]0,1[ then it also holds for allδ in ]0,1[ and we estimate the functionC=C(δ). We show that this property holds iff the “volume ratio” of the finite dimensional subspaces ofX are uniformly bounded. We also show that (althoughX can have this property without being of cotype 2)L2(X) possesses this property iffX if of cotype 2. In the last part of the paper, we study theK-convex spaces which have a dual with the above property and we relate it to a certain extension property.

Unconditional and symmetric sets inn-dimensional normed spaces

Isoperimetric inequalities are used to obtain measure estimates on almost constancy sets of functions on product spaces. These are applied to produce almost unconditional or symmetric block sequences from given sequences. Their length, which is (logn)1/2 in the general case, improves tona where a cotype condition is imposed or when the given sequences arep-type attaining for somep<2. In thep-type attaining case, block sequences (1+ε)-equivalent to the unit vector basis oflpmcan be obtained when log logm ∼ log logn.

Coordinate density of sets of vectors

Abstract Let no = 0,n1 ⩾1…,n3 ⩾1 be given natural numbers, J i = ∑ t=0 0−1 n i +1,…, ∑ t=0 t n t (i=J i ,…,s) and Π t=0 s E n a = (x( t) ,.…, x( 1 ): n= ∑ i=1 s n 1 and if rϵ J i , then x( u )ϵ{0,…, q i −J} A set R⊆∏i3 = 1 Eaji2 is said to be (m1, …,m3)-dense (1⩽m1⩽n) if there exist I1⊆J1 such that |L1| = m6 (i = 1,…,s) and |L(I)(R)| = ∏I5 = qlm1 where P(I)(R) is the projection of R on the coordinate axes whose indices lie in I = ∪i5 = jL1. In this paper we establish necessary and sufficient conditions for an arbitrary set R⊆∏51 = 1 Ea1n1 with given |R| to be (m1,…,m3)-dense.