Isomorphically Polyhedral Banach Spaces

We prove two theorems giving su±cient conditions for a Banach space to be isomorphically polyhedral.Доказаны две теоремы, дающие достаточные условия для того, чтобы банахово пространство было изоморфически многогранным

Almost flat locally finite coverings of the sphere

For any subset A of the unit sphere of a Banach space X and for [0,2) the notion of -flatness is introduced as a measure of non-flatness of A. For any positive , construction of locally finite tilings of the unit sphere by -flat sets is carried out under suitable -renormings of X in a quite general context; moreover, a characterization of spaces having separable dual is provided in terms of the existence of such tilings. Finally, relationships between the possibility of getting such tilings of the unit sphere in the given norm and smoothness properties of the norm are discussed

Covering a Banach space

A well-known theorem by H. Corson states that if a Banach space admits a locally finite covering by bounded closed convex subsets, then it contains no infinite-dimensional reflexive subspace. We strengthen this result proving that if an infinite-dimensional Banach space admits a locally finite covering by bounded -closed subsets, then it is -saturated, thus answering a question posed by V. Klee concerning locally finite coverings of spaces. Moreover, we provide information about massiveness of the set of singular points in (PC) spaces

On Tauberian and co-Tauberian operators

We show that a Banach space X has an infinite dimensional reflexive subspace (quotient) if and only if there exist a Banach space Z and a non- isomorphic one-to-one (dense range) Tauberian (co-Tauberian) operator form X to Z (Z to Z). We also give necessary and sufficient condition for the existence of a Tauberian operator from a separable Banach space to c0 which in turn generalizes a result of Johnson and Rosenthal. Another application of our result shows that if X** is separable, then there exists a renorming of X for which, X is essentially the only subspace contained in the set of norm attaining functionals on X*.Partially supported by the Institute for Advanced Studies in Mathematics at Ben-Gurion University of the Negev and by Israel Science Foundation, Grant No. 139/03.peerReviewe

Coverings of Banach spaces: beyond the Corson theorem

A well known result due to H. Corson has been recently improved by the authors. In its final form it essentially reads as follows: for any covering $\tau$ by closed bounded convex subsets of any Banach space $X$ containing a separable infinite-dimensional dual space, a (algebraically) finite-dimensional compact set $C$ can always be found that meets infinitely many members of $\tau$. In the present paper we investigate how small the dimension of this compact set can be, in the case the members of $\tau$ are closed bounded convex bodies satisfying general conditions of rotundity or smoothness type. In particular, such a compact set turns out to be a segment whenever the members of $\tau$ are rotund or smooth bodies in the usual sense

Corrigendum to "Best approximation in polyhedral Banach spaces" [J. Approx. Theory 163 (11) (2011, 1748-1771)]

The present note is a corrigendum to the paper "Best approximation in polyhedral Banach spaces", J. Approx. Theory 163 (2011) 1748-1771