On strong proximinality in normed linear spaces

The paper deals with strong proximinality in normed linear spaces. It is proved that in a compactly locally uniformly rotund Banach space, proximinality, strong proximinality, weak approximative compactness and approximative compactness are all equivalent for closed convex sets. How strong proximinality can be transmitted to and from quotient spaces has also been discussed.

A New Characterization ofk-Uniformly Rotund Banach Space

A new characterization ofk-uniformly rotund Banach space with1<P<+∞is given. Moreover, a corresponding result in the locallyk-uniformly rotund Banach space with1<P<+∞is given.

Some questions about rotundity and renormings in Banach spaces

In this paper, we show some results involving classical geometric concepts. For example, we characterize rotundity and Efimov-Stechkin property by mean of faces of the unit ball. Also, we prove that every almost locally uniformly rotund Banach space is locally uniformly rotund if its norm is Fréchet differentiable. Finally, we also provide some theorems in which we characterize the (strongly) exposed points of the unit ball using renormings.

On the weak uniform rotundity of Banach spaces

We prove that ifXi,i=1,2,…,are Banach spaces that are weak* uniformly rotund, then theirlpproduct space(p>1)is weak* uniformly rotund, and for any weak or weak* uniformly rotund Banach space, its quotient space is also weak or weak* uniformly rotund, respectively.

Asplund spaces and a variant of weak uniform rotundity

We introduce a property formally weaker than weak uniform rotundity, which we call equatorial weak uniform rotundity. We show that an equatorially weakly uniformly rotund norm need not be weakly locally uniformly rotund. Nevertheless, we show that an equatorially weakly uniformly rotund Banach space is an Asplund space.

K-Uniform Rotundity of Sequence Orlicz Spaces

This paper presents a criterion of KUR for sequence Orlicź spaces with Luxemburg's norm. The result also indicates that for any integer k ≥ 1, there exists a k + 1-uniformly rotund Banach space not being k-uniformly rotund.