Olesia Zavarzina

V. N. Karazin Kharkiv National University · Department of Pure Mathematics

The Strong Law of Large Numbers (SLLN) for random variables or random vectors with different mathematical expectations easily reduces by means of shifts to SLLN for random variables or random vectors whose mathematical expectations are equal to zero. The situation changes for random sets, where shifts cannot reduce sets of more than one point to th...

The study deals with plastic and non-plastic sub-spaces A of the real-line ℝ with the usual Euclidean metric d . It investigates non-expansive bijections, proves properties of such maps, and demonstrates their relevance by hands of examples. Finally, it is shown that the plasticity property of a sub-space A contains at least two complementary quest...

It is known that if any function acting from precompact metric space to itself increases the distance between some pair of points then it must decrease distance between some other pair of points. We show that this is not the case for quasi-metric spaces. After that, we present some sufficient conditions under which the previous property holds true...

We address pairs $(X, Y)$ of metric spaces with the following property: for every mapping $f: X \to Y$ the existence of points $x, y \in X$ with $d(f(x),f(y)) > d(x,y)$ implies the existence of $\widetilde{x}, \widetilde{y}\in X$ for which $d(f(\widetilde{x}),f(\widetilde{y})) < d(\widetilde{x},\widetilde{y})$. We give sufficient conditions for thi...

It is known that if any function acting from precompact metric space to itself increases the distance between some pair of points then it must decrease distance between some other pair of points. We show that this is not the case for quasi-metric spaces. After that, we present some sufficient conditions under which the previous property holds true...

This work is aimed to describe linearly expand-contract plastic ellipsoids given via quadratic form of a bounded positively defined self-adjoint operator in terms of its spectrum.Let $Y$ be a metric space and $F\colon Y\to Y$ be a map. $F$ is called non-expansive if it does not increase distance between points of the space $Y$. We say that a subset...

We prove that Banach spaces ℓ1 ⊕2 R and X ⊕∞ Y , with strictly convex X and Y , have plastic unit balls (we call a metric space plastic if every non-expansive bijection from this space onto itself is an isometry).

This work is aimed to describe linear expand-contract plastic ellipsoids given via quadratic form of a bounded positively defined self-adjoint operator in terms of its spectrum.

We prove that Banach spaces $\ell_1\oplus_2\mathbb{R}$ and $X\oplus_\infty Y$, with strictly convex $X$ and $Y$, have plastic unit balls (we call a metric space plastic if every non-expansive bijection from this space onto itself is an isometry).

************************** Springer Nature SharedIt link: **************************** https://rdcu.be/b3Cfp *************************************ABSTRACT***************************************** We study geometric properties of GL-spaces. We demonstrate that every finite-dimensional GL-space is polyhedral; that in dimension 2 there are only two,...

The paper is aimed to establish the interdependence between linear expand-contract plasticity of an ellipsoid in a separable Hilbert spaces and properties of the set of its semi-axes.

We extend the result of B. Cascales et al. about expand-contract plasticity of the unit ball of strictly convex Banach space to those spaces whose unit sphere is the union of all its finite-dimensional polyhedral extreme subsets. We also extend the definition of expand-contract plasticity to uniform spaces and generalize the theorem on expand-contr...

We extend the result of B. Cascales at al. about expand-contract plasticity of the unit ball of strictly convex Banach space to those spaces whose unit ball is the union of all its finite-dimensional polyhedral extreme subsets. We also extend the definition of expand-contract plasticity to uniform spaces and generalize the theorem on expand-contrac...

Extending recent results by Cascales, Kadets, Orihuela and Wingler (2016), Kadets and Zavarzina (2017), and Zavarzina (2017) we demonstrate that for every Banach space $X$ and every collection $Z_i, i\in I$ of strictly convex Banach spaces every non-expansive bijection from the unit ball of $X$ to the unit ball of sum of $Z_i$ by $\ell_1$ is an iso...

It is known that if $M$ is a finite-dimensional Banach space, or a strictly convex space, or the space $\ell_1$, then every non-expansive bijection $F: B_M \to B_M$ is an isometry. We extend these results to non-expansive bijections $F: B_E \to B_M$ between unit balls of two different Banach spaces. Namely, if $E$ is an arbitrary Banach space and $...

In the recent paper by Cascales, Kadets, Orihuela and Wingler it is shown that for every strictly convex Banach space $X$ every non-expansive bijection $F: B_X \to B_X$ is an isometry. We extend this result to the space $\ell_1$, which is not strictly convex. http://vestnik-math.univer.kharkov.ua/Vestnik-KhNU-83-2016-kadets.pdf