Comparison of Some Distances Between Sums of ℓ n p Spaces

In this paper, we study the classical, modified, and weak Banach–Mazur distances between sums of _p ⁿ spaces. We explicitly calculate the classical and weak Banach–Mazur distances between sums of _p ⁿ spaces and establish bounds for the ratios of these distances.     

John's Theorem for an Arbitrary Pair of Convex Bodies

We provide a generalization of John's representation of the identity for the maximal volume position of L inside K, where K and L are arbitrary smooth convex bodies in ⁿ . From this representation we obtain Banach–Mazur distance and volume ratio estimates.     

Generic Banach spaces and generic simplexes

We give a systematic study of certain class of generic Banach spaces. We show that they distinguish between an array of different properties related to smoothness of equivalent norms such as for example the Mazur intersection property or the existence of convex sets supported by all of their points. We also examine the dual constructions of generic Choquet simplexes with extra requirements such as for example those of Poulsen and Bauer asking that the set of extremal points is dense or closed, respectively.     

On the Casson invariant of homology 3-spheres of Mazur type

We compute the Casson invariant for some integral homology 3-spheres. We also show that for every integer n, there exists an integral homology 3-sphere of Mazur type with the Casson invariant 2n.     

Allometry constants of finite-dimensional spaces: theory and computations

We describe the computations of some intrinsic constants associated to an n-dimensional normed space , namely the N-th “allometry” constants

Mazur Sets in Normed Spaces

In this paper we investigate different questions concerning Mazur sets in normed spaces, which point out the close connections between geometric functional analysis and discrete geometry. Motivated by a result of Chen and Lin, we study the relationship between Mazur disks and weak* denting points of the dual unit ball. We prove that the only Mazur sets of the spaces l₁ⁿ are points and closed balls. Finally, a new stability property for the family of all sets which are intersections of closed balls is found.     

Integrability of Analytic Almost Complex Structures on Banach Manifolds

We prove that the classical integrability condition for almost complex structures on finite-dimensional smooth manifolds also works in infinite dimensions in the case of almost complex structures that are real analytic on real analytic Banach manifolds. With this result at hand, we extend some known results concerning existence of invariant complex structures on homogeneous spaces of Banach–Lie groups. By way of illustration, we construct the complex flag manifolds associated with unital C^*-algebras.Mathematics Subject Classifications (2000): primary 32Q60; secondary 53C15, 58B12.     

关于C-序列空间与Mazur空间的进一步研究

本文深入讨论 C-序列空间与 Mazur空间的性质 ,对二者进行了比较并分别证明了其遗传性 .     

Banach Spaces Which Are Far from all Lattices

We consider n-dimensional real Banach spaces X which are far, in the Banach–Mazur distance, from all complemented subspaces of all Banach lattices. We show that this is related to the volume ratio values of X with respect to ellipsoids and to zonoids.     

发汗冷却控制系统最优控制元的存在唯一性

通过应用泛函分析中一个重要的Mazur定理,证出了水发汗冷却控制系统在自反的Banach空间中的最优控制元的存在唯一性。     

On the connectedness of isomorphism classes

This paper is concerned with some topological properties of a notion of distance between normed linear spaces which is due to Banach and Mazur [1], We are interested in a pseudometric space associated naturally with this notion of distance, and our main result is that this pseudometric space is always pathwise connected.The results in this paper were taken from the author's doctoral dissertation written at the University of Maryland under the direction of Robert Whitley. The research was partially supported by the N.S.F.     

Mazur交性质

主要考虑关于Mazur交性质的问题．我们通过赋范半群的对偶,给出了一个赋范空间只有平凡Mazur集白々等价刻画．另外,我们还讨论乘积空间对Mazur交性质的继承性,对Sersouri和Roy的结论作了一定的改进,将范数推广至更一般的情形．     

Gelfand–Mazur Theorems in normed algebras: A survey

The Gelfand–Mazur Theorem, a very basic theorem in the theory of Banach algebras states that: (Real version) Every real normed division algebra is isomorphic to the algebra of all real numbers $\mathbb{R}$, the complex numbers $\mathbb{C}$ or the quaternions $\mathbb{H}$; (Complex version) Every complex normed division algebra is isometrically isomorphic to $\mathbb{C}$. This theorem has undergone a large number of generalizations. We present a survey of these generalizations and also discuss some closely related unsettled issues.     

On the Asymmetry Constant of a Body with Few Vertices

In this note, we show that a nondegenerated polytope in IR ⁿ with n + k, 1 k < n, vertices is far from any symmetric body. We provide the asymptotically sharp estimates for the asymmetry constant of such polytopes.     

Approximation of Convex Bodies by Centrally Symmetric Bodies

We present an analog of the well-known theorem of F. John about the ellipsoid of maximal volume contained in a convex body. Let C be a convex body and let D be a centrally symmetric convex body in the Euclidean d-space. We prove that if D is an affine image of D of maximal possible volume contained in C, then C a subset of the homothetic copy of D with the ratio 2d-1 and the homothety center in the center of D. The ratio 2d-1 cannot be lessened as a simple example shows.     

Mazur intersection property for Asplund spaces

The main result of the present note states that it is consistent with the ZFC axioms of set theory (relying on Martin's Maximum MM axiom), that every Asplund space of density character ω₁ has a renorming with the Mazur intersection property. Combined with the previous result of Jiménez and Moreno (based upon the work of Kunen under the continuum hypothesis) we obtain that the MIP renormability of Asplund spaces of density ω₁ is undecidable in ZFC.     

ON THE UNIFORMLY FRECHETDIFFERENTIABILITY OF NORMS

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Continuity of the Restriction of C 0-Semigroups to Invariant Banach Subspaces

A linear semigroup in a Banach space induces a linear semigroup on a Banach space that can be continuously embedded in the former such that its image is invariant. This restriction need not be strongly continuous, although the original semigroup is strongly continuous. We show that norm or weak compactness of partial orbits is a necessary and sufficient condition for strong continuity of the restriction of a C₀-semigroup. We then show that if the embedded Banach space is reflexive and the norms of the restricted semigroup operators are bounded near the initial time, then the restricted semigroup is strongly continuous.     

Metrics and tangent cones of uniformly regular Carnot—Carathéodory spaces

Given a uniformly regular Carnot—Carathéodory space, we prove equivalence of the quasimetrics generated by various bases of vector fields which agree with filtration of the space. We prove a theorem on a nilpotent tangent cone for a uniformly regular Carnot—Carathéodory space furnished with quasimetrics. As a consequence, we obtain a theorem on isomorphism of nilpotent tangent cones defined at a common distinguished point.     

赋序列范数的矢值序列空间ss(E)的强暴露性和暴露性

刻划了赋序列范数的矢值B anach序列空间ss(E)的强暴露性和暴露性,给出了它们的判据.