Related papers: On Banach Spaces with the Helly Approximation Prop…

We study delta-points in Banach spaces $h_{\mathcal{A},p}$ generated by adequate families $\mathcal A$ where $1 \le p < \infty$. In the case the familiy $\mathcal A$ is regular and $p=1,$ these spaces are known as combinatorial Banach…

Let $\mathbb{F}$ be a fixed field and let $X$ be a simplicial complex on the vertex set $V$. The Leray number $L(X;\mathbb{F})$ is the minimal $d$ such that for all $i \geq d$ and $S \subset V$, the induced complex $X[S]$ satisfies…

We study $M$-ideals of compact operators by means of the property~$(M)$ introduced in \cite{Kal-M}. Our main result states for a separable Banach space $X$ that the space of compact operators on $X$ is an $M$-ideal in the space of bounded…

We use the notion of $\A$-compact sets, which are determined by a Banach operator ideal $\A$, to show that most classic results of certain approximation properties and several Banach operator ideals can be systematically studied under this…

Let $X$ be a Hausdorff topological vector space, $X^*$ its topological dual and $Z$ a subset of $X^*$. In this paper, we establish some results concerning the $\sigma(X,Z)$-approximate fixed point property for bounded, closed convex subsets…

We introduce and investigate a quantitative version of Steinhaus' property $(S)$ for Banach spaces, called the uniform property $(S)$. A Banach space $X$ is said to have uniform $(S)$ if for every pair of distinct unit vectors $x,y\in X$…

For Banach spaces $X,Y,$ we consider a distance problem in the space of bounded linear operators $\mathcal{L}(X,Y).$ Motivated by a recent paper \cite{RAO21}, we obtain sufficient conditions so that for a compact operator…

This paper will generalize what may be termed the "geometric duality theory" of real pre-ordered Banach spaces which relates geometric properties of a closed cone in a real Banach space, to geometric properties of the dual cone in the dual…

We study the computational complexity of determining the Hausdorff distance of two polytopes given in halfspace- or vertex-presentation in arbitrary dimension. Subsequently, a matching problem is investigated where a convex body is allowed…

In this paper we investigate the property (HLUR), a generalisation of (LUR) property of a Banach space. A Banach space having the property (HLUR) is called an HLUR space. We characterise (HLUR) property with the help of known geometric…

We study the best coapproximation problem in Banach spaces, by using Birkhoff-James orthogonality techniques. We introduce two special types of subspaces, christened the anti-coproximinal subspaces and the strongly anti-coproximinal…

Let $\mathcal{P}$ be a class of Banach spaces and let $T=\{T_\alpha\}_{\alpha\in A}$ be a set of metric spaces. We say that $T$ is a set of {\it test-spaces} for $\mathcal{P}$ if the following two conditions are equivalent: (1)…

We prove that the kernel of a quotient operator from an $\mathcal L_1$-space onto a Banach space $X$ with the Bounded Approximation Property (BAP) has the BAP. This completes earlier results of Lusky --case $\ell_1$-- and Figiel, Johnson…

We intend to study the uniqueness of the Hahn-Banach extensions of linear functionals on a subspace in locally convex spaces. Various characterizations are derived when a subspace $Y$ has an analogous version of property-U (introduced by…

We show that for every $1<n<\infty$, there exits a Banach space $X_n$ containing proximinal subspaces of codimension $n$ but no proximinal finite codimensional subspaces of higher codimension. Moreover, the set of norm-attaining functionals…

An important consequence of the Hahn-Banach Theorem says that on any locally convex Hausdorff topological space $X$, there are sufficiently many continuous linear functionals to separate points of $X$. In the paper, we establish a `local'…

We present some extensions of classical results that involve elements of the dual of Banach spaces, such as Bishop-Phelp's theorem and James' compactness theorem, but restricting to sets of functionals determined by geometrical properties.…

We prove that the dual of an M ideal of a Banach space inherits all the versions of $w^*$ diameter two properties. We give a counter example to show that the converse is not true. We use these results to explore these properties in $C(K)$…

We present some results related to Hahn-Banach extension theorem for linear operators on asymmetric normed spaces. L. Nachbin, Trans. Amer. Math. Soc. 68 (1950), proved that a Banach space has the extension property for linear operators (a…

Suppose that $X$ is a Banach space. We will show that $X$ does not contain a copy of $c_0$ if and only if for each series which is not unconditionally convergent in $X$ respective sets coding all bounded subseries and rearrangements are…