A proportional Dvoretzky-Rogers factorization result

If is an -dimensional normed space and , there exists , such that the formal identity can be written as , with . This is proved as a consequence of a Sauer-Shelah type theorem for ellipsoids.

     

The Banach-Mazur distance between symmetric spaces

We show that the Banach-Mazur distance betweenN-dimensional symmetric spacesE andF satisfies , wherec is a numerical constant. IfE is a symmetric space, then max (M ⁽²⁾(E),M ₍₂₎(E)), whereM ⁽²⁾(E) (resp.M ₍₂₎(E)) denotes the 2-convexity (resp. the 2-concavity) constant ofE. We also give an example of a spaceF with an 1-unconditional basis and enough symmetries that satisfiesd(F, l ₂ ^dimF )=M ⁽²⁾(F)M ₍₂₎(F). Partially supported by NSF Grant MCS-8201044.     

Generalization of some classical results to the case of the modified Banach-Mazur distance

The paper is devoted to generalization of some classical results concerning the Banach-Mazur distance to the modified Banach-Mazur distance. We establish the existense of a space that is uniformly distant in the modified Banach-Mazur distance from all spaces with a small basis constant and of a space that is distant in the modified metric from all spaces admitting a complex structure. The existense of a real space that admits two complex structures and is distant in the sense of the complex modified distance is established. The existense of a space having large generalized volume ratio with all of its subspaces of proportional dimension is proved. Bibliography: 10 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 333, 2006, pp. 17–32.     

Daisy cubes and distance cube polynomial

Let $X\subseteq {\{0,1\}}^n$. The daisy cube $Q_n\left(X\right)$ is introduced as the subgraph of $Q_n$ induced by the union of the intervals $I(x,0^n)$ over all $x\in X$. Daisy cubes are partial cubes that include Fibonacci cubes, Lucas cubes, and bipartite wheels. If $u$ is a vertex of a graph $G$, then the distance cube polynomial $D_{G,u}(x,y)$ is introduced as the bivariate polynomial that counts the number of induced subgraphs isomorphic to $Q_k$ at a given distance from the vertex $u$. It is proved that if $G$ is a daisy cube, then $D_{G,0^n}(x,y)=C_G(x+y-1)$, where $C_G\left(x\right)$ is the previously investigated cube polynomial of $G$. It is also proved that if $G$ is a daisy cube, then $D_{G,u}(x,-x)=1$ holds for every vertex $u$ in $G$.     

A proof of the Dvoretzky-Rogers theorem

We give a new proof of the famous Dvoretzky-Rogers theorem ([2], Theorem 1), according to which a Banach spaceE is finite-dimensional if every unconditionally convergent series inE is absolutely convergent.     

On the diameter of the Banach-Mazur set

On every subspace of l _∞(ℕ) which contains an uncountable ω-independent set, we construct equivalent norms whose Banach-Mazur distance is as large as required. Under Martin’s Maximum Axiom (MM), it follows that the Banach-Mazur diameter of the set of equivalent norms on every infinite-dimensional subspace of l _∞(ℕ) is infinite. This provides a partial answer to a question asked by Johnson and Odell.     

The multicriteria big cube small cube method

In this paper we propose the big cube small cube (BCSC) technique for multicriteria optimization problems. The output of our algorithm results in a set which consists of epsilon efficient solutions and which contains all efficient solutions.     

Topological spaces related to the Banach-Mazur game and the generic well-posedness of optimization problems

The existence of special kind of winning strategies in the Banach-Mazur game in a completely regular topological spaceX is shown to be equivalent to generic stability properties of optimization problems generated by the continuous bounded real-valued functions inX.Research partially supported by the National Foundation for Scientific Research at the Bulgarian Ministry of Education and Science under Grant Number MM-408/94.     

West's problem on equivariant hyperspaces and Banach-Mazur compacta

Let be a compact Lie group, a metric -space, and the hyperspace of all nonempty compact subsets of endowed with the Hausdorff metric topology and with the induced action of . We prove that the following three assertions are equivalent: (a) is locally continuum-connected (resp., connected and locally continuum-connected); (b) is a -ANR (resp., a -AR); (c) is an ANR (resp., an AR). This is applied to show that is an ANR (resp., an AR) for each compact (resp., connected) Lie group . If is a finite group, then is a Hilbert cube whenever is a nondegenerate Peano continuum. Let be the hyperspace of all centrally symmetric, compact, convex bodies , , for which the ordinary Euclidean unit ball is the ellipsoid of minimal volume containing , and let be the complement of the unique -fixed point in . We prove that: (1) for each closed subgroup , is a Hilbert cube manifold; (2) for each closed subgroup acting non-transitively on , the -orbit space and the -fixed point set are Hilbert cubes. As an application we establish new topological models for tha Banach-Mazur compacta and prove that and have the same -homotopy type.

     

Banach-Mazur Distances and Projections on Random Subgaussian Polytopes

We consider polytopes in that are generated by N vectors in whose coordinates are independent subgaussian random variables. (A particular case of such polytopes are symmetric random polytopes generated by N independent vertices of the unit cube.) We show that for a random pair of such polytopes the Banach-Mazur distance between them is essentially of a maximal order n. This result is an analogue of the well-known Gluskin's result for spherical vectors. We also study the norms of projections on such polytopes and prove an analogue of Gluskin's and Szarek's results on basis constants. The proofs are based on a version of "small ball" estimates for linear images of random subgaussian vectors.     

The application of Padeapproximants to Wiener-Hopf factorization

The key step in the solution of a Wiener–Hopf equation is the decomposition of the Fourier transform of the kernel,which is a function of a complex variable, say, into a productof two terms. One is singularity and zero free in an upperregion of the -plane, and the other singularity and zero freein an overlapping lower region. Each product factor can beexpressed in terms of a Cauchy-type integral formula, but thisform presents difficulties due to the speed of its evaluationand numerical problems caused by singularities near the integrationcontour. Other representations are available in special cases,for instance an infinite product form for meromorphic functions,but not in general. To overcome these problems, several approximatemethods for decomposing the transformed kernels have been suggested.However, whilst these offer simple explicit expressions, theirforms tend to have been derived in an ad hoc fashion and todate have only mediocre accuracy (of order one per cent orso). A new method for approximating Wiener–Hopf kernelsis offered in this article which employs Padéapproximants.These have the advantage of offering very simple approximatefactors of Fourier transformed kernels which are found to beextremely accurate for modest computational effort. Further,the derivation of the factors is algorithmic and thereforerequires little effort, and the Padénumber is a convenientparameter with which to reduce errors to within set targetvalues. The paper demonstrates the efficacy of the approachon several model kernels, and numerical results presented hereinconfirm theoretical predictions regarding convergence to theexact results, etc. The relationship between the present methodand earlier approximate schemes is discussed. Received 7 February, 1998. Revised 18 February, 1999.     

The factorization approach to large-scale linear programming

A unifying concept for large-scale linear programming is developed. This approach, calledfactorization, allows one to isolate the effect of different types of constraints and variables in the algebraic representation of the tableau. Two different factorizations based on a double representation of the tableau are developed. These factorizations are applied to obtain the essential structure of efficient algorithms for generalized upper bounding, coupled block-diagonal problems, set partitioning LPs, minimum cost network flows, and other classes of problems.This research was supported in part by the National Science Foundation and the Office of Naval Research. An earlier version of this paper was presented at the Eighth International Symposium on Mathematical Programming, Stanford University, August 1973.     

The multidimensional cube recurrence

We introduce a recurrence which we term the multidimensional cube recurrence, generalizing the octahedron recurrence studied by Propp, Fomin and Zelevinsky, Speyer, and Fock and Goncharov and the three-dimensional cube recurrence studied by Fomin and Zelevinsky, and Carroll and Speyer. The states of this recurrence are indexed by tilings of a polygon with rhombi, and the variables in the recurrence are indexed by vertices of these tilings. We travel from one state of the recurrence to another by performing elementary flips. We show that the values of the recurrence are independent of the order in which we perform the flips; this proof involves nontrivial combinatorial results about rhombus tilings which may be of independent interest. We then show that the multidimensional cube recurrence exhibits the Laurent phenomenon - any variable is given by a Laurent polynomial in the other variables. We recognize a special case of the multidimensional cube recurrence as giving explicit equations for the isotropic Grassmannians IG(n−1,2n). Finally, we describe a tropical version of the multidimensional cube recurrence and show that, like the tropical octahedron recurrence, it propagates certain linear inequalities.     

About the prohorov distance between the uniform distribution over the unit cube in R dand its empirical measure

Summary We compute the almost sure order of convergence of the Prokhorov distance between the uniform distribution P over [0, 1] ^d and the empirical measure associated with n independent observations with (common) distribution P. We show that this order of convergence is n ^-1/d up to a power of log(n). This result extends to the case where the observations are weakly dependent.