Related papers: Farthest Point Problem and Partial Statistical Con…

A Banach space has the weak fixed point property if its dual space has a weak$^*$ sequentially compact unit ball and the dual space satisfies the weak$^*$ uniform Kadec-Klee property; and it has the \fpp if there exists $\epsilon>0$ such…

Consider the continuum of points along the edges of a network, i.e., a connected, undirected graph with positive edge weights. We measure the distance between these points in terms of the weighted shortest path distance, called the network…

A remarkable theorem of R. C. James is the following: suppose that $X$ is a Banach space and $C \subseteq X$ is a norm bounded, closed and convex set such that every linear functional $x^* \in X^*$ attains its supremum on $C$; then $C$ is a…

This paper establishes limit theorems and quantitative statistical stability for a class of piecewise partially hyperbolic maps that are not necessarily continuous nor locally invertible. By employing a flexible functional-analytic…

$C_p(X)$ denotes the space of continuous real-valued functions on a Tychonoff space $X$ endowed with the topology of pointwise convergence. A Banach space $E$ equipped with the weak topology is denoted by $E_{w}$. It is unknown whether…

We study the problem of minimizing the supremum norm by monic polynomials with integer coefficients. Let ${\M}_n({\Z})$ denote the monic polynomials of degree $n$ with integer coefficients. A {\it monic integer Chebyshev polynomial} $M_n…

We show that if the Banach-Mazur distance between an n-dimensional normed space X and ell infinity is at most 3/2, then there exist n+1 equidistant points in X. By a well-known result of Alon and Milman, this implies that an arbitrary…

We study some properties of the randomized series and their applications to the geometric structure of Banach spaces. For $n\ge 2$ and $1<p<\infty$, it is shown that $\ell_\infty^n$ is representable in a Banach space $X$ if and only if it…

A separable Banach space $X$ is said to be finitely determined if for each separable space $Y$ such that $X$ is finitely representable (f.r.) in $Y$ and $Y$ is f.r. in $X$ then $Y$ is isometric to $X$. We provide a direct proof (without…

A metric space $\mathbf{X}$ is called densely complete if there exists a dense set $D$ in $\mathbf{X}$ such that every Cauchy sequence of points of $D $ converges in $\mathbf{X}$. One of the main aims of this work is to prove that the…

For any finite point set in $D$-dimensional space equipped with the 1-norm, we present random linear embeddings to $k$-dimensional space, with a new metric, having the following properties. For any pair of points from the point set that are…

We investigate strongly separately continuous functions on a product of topological spaces and prove that if $X$ is a countable product of real lines, then there exists a strongly separately continuous function $f:X\to\mathbb R$ which is…

Let C be a nonempty closed convex subset of a Banach space $E$. In this paper we introduce a new definition for variational inequality V I (C, B) on E that generalizes the analogue definition on Hilbert spaces. We generalize (u,…

We consider maps defined on the interior of a normal, closed cone in a real Banach space that are nonexpansive with respect to Thompson's metric. With mild compactness assumptions, we prove that the Krasnoselskii iterates of such maps…

In the main result of the paper we extend Rosenthal's characterization of Banach spaces with the Schur property by showing that for a quasi-complete locally convex space $E$ whose separable bounded sets are metrizable the following…

Let $\mathcal{P}$ be a simple polygon with $m$ vertices and let $P$ be a set of $n$ points inside $\mathcal{P}$. We prove that there exists, for any $\varepsilon>0$, a set $\mathcal{C} \subset P$ of size $O(1/\varepsilon^2)$ such that the…

We show that the direct sum of Banach spaces $X_{1},..., X_{r}$ with a strictly monotone norm has the weak fixed point property for nonexpansive mappings whenever $M(X_{i})>1$ for each $i=1,...,r$. In particular, $(X_{1} \oplus ... \oplus…

We prove that for every function $f:X\to Y$, where $X$ is a separable Banach space and $Y$ is a Banach space with RNP, there exists a set $A\in\tilde\mcA$ such that $f$ is Gateaux differentiable at all $x\in S(f)\setminus A$, where $S(f)$…

Let $\{Y_i\}_{i=1}^{\infty}$ be a stationary reversible Markov chain with state space $[N]$, let $(X, \| \cdot \|)$ be a real-valued Banach space and let $f_1, \ldots, f_n: [N] \rightarrow X$ be functions with mean $0$ such that $\|f_i(v)\|…

Let $E$ be one of the spaces $C(K)$ and $L_1$, $F$ be an arbitrary Banach space, $p>1,$ and $(X,\sigma)$ be a space with a finite measure. We prove that $E$ is isometric to a subspace of the Lebesgue-Bochner space $L_p(X;F)$ only if $E$ is…