Related papers: Some remarks on contractive and existence sets

Given a closed set $C$ in a Banach space $(X, \|\cdot\|)$, a point $x\in X$ is said to have a nearest point in $C$ if there exists $z\in C$ such that $d_C(x) =\|x-z\|$, where $d_C$ is the distance of $x$ from $C$. We shortly survey the…

Let ${\bf x}=(x_n)_n$ be a sequence in a Banach space. A set $A\subseteq \mathbb{N}$ is perfectly bounded, if there is $M$ such that $\|\sum_{n\in F}x_n\|\leq M$ for every finite $F\subseteq A$. The collection $B({\bf x})$ of all perfectly…

In this paper, we study the existence of the random approximations and fixed points for random almost lower semicontinuous operators defined on finite dimensional Banach spaces, which in addition, are condensing or 1-set-contractive. Our…

In this article we treat a notion of continuity for a multi-valued function F and we compute the descriptive set-theoretic complexity of the set of all x for which F is continuous at x. We give conditions under which the latter set is…

A set $X$ endowed with a coarse structure is called ballean or coarse space. For a ballean $(X, \mathcal{E})$, we say that two subsets $A$, $B$ of $X$ are close (linked) if there exists an entourage $E\in \mathcal{E}$ such that $A\subseteq…

We study some questions concerning the structure of the set of spreading models of a separable infinite-dimensional Banach space $X$. In particular we give an example of a reflexive $X$ so that all spreading models of $X$ contain $\ell_1$…

We continue to study (strong) property-$(R_1)$ in Banach spaces. As discussed by Pai \& Nowroji in [{\it On restricted centers of sets}, J. Approx. Theory, {\bf 66}(2), 170--189 (1991)], this study corresponds to a triplet…

Under $\mathrm{ZF}$, we show that the statement that every subset of every $\mathbb{R}$-vector space has a maximal convex subset is equivalent to the Axiom of Choice. We also study the strength of the same statement restricted to some…

We establish a general criterion for the existence of convex sets of fixed shape as, e.g., balls of a given radius, of maximal probability on Banach spaces. We also provide counterexamples showing that their existence my fail even in some…

Given a compact metric space $X$ and an upper semicontinuous function $F\colon X \to 2^X$, we explore the dynamic system $(X,F)$. In this study, we introduce new concepts, demonstrate various results, and provide numerous examples. In…

The main results of the paper: {\bf (1)} The dual Banach space $X^*$ contains a linear subspace $A\subset X^*$ such that the set $A^{(1)}$ of all limits of weak$^*$ convergent bounded nets in $A$ is a proper norm-dense subset of $X^*$ if…

Let $(G,+)$ be an Abelian group. Given $h\in \mathbb{Z}^+$, a non-empty subset $A$ of $G$ is called an $S_h$-set if all the sums of $h$ distinct elements of $A$ are different. We extend the concept of $S_h$-set to a more general context in…

We work in set-theory without choice ZF. Denoting by AC(N) the countable axiom of choice, we show in ZF+AC(N) that the closed unit ball of a uniformly convex Banach space is compact in the convex topology (an alternative to the weak…

Given a compact basic semi-algebraic set $K\subset R^n\times R^m$, a simple set $B$ (box or ellipsoid), and some semi-algebraic function $f$, we consider sets defined with quantifiers, of the form $R_f:=\{x\in B: \mbox{$f(x,y)\leq 0$ for…

If $(X,d)$ is a metric space then the map $f\colon X\to X$ is defined to be a weak contraction if $d(f(x),f(y))<d(x,y)$ for all $x,y\in X$, $x\neq y$. We determine the simplest non-closed sets $X\subseteq \mathbb{R}^n$ in the sense of…

The classical theorem of Bishop-Phelps asserts that, for a Banach space X, the norm-achieving functionals in X* are dense in X*. Bela Bollobas's extension of the theorem gives a quantitative description of just how dense the norm-achieving…

It is shown that any Banach space X of sufficiently large density contains an (infinite) unconditional sequence and a separable quotient. If a density of X is a weakly compact cardinal, then X contains an unconditional sequence of…

A wide new class of subsets of a Banach space $X$ named coarse $p$-limited sets ($ 1\leq p < \infty$) is introduced by considering weak* $p$-summable sequences in $X'$ instead of weak* null sequences. We study its basic properties and…

For a nonempty topological space X, the ring of all real-valued functions on $X$ with pointwise addition and multiplication is denoted by $F(X)$ and continuous members of $F(X)$ is denoted by $C(X)$. Let $A(X)$ be a subring of $F(X)$ and…

A topological space $X$ is called a topological fractal if $X=\bigcup_{f\in\mathcal F}f(X)$ for a finite system $\mathcal F$ of continuous self-maps of $X$, which is topologically contracting in the sense that for every open cover $\mathcal…