Related papers: Compactness and D-boundedness in Menger's 2-probab…

In the present paper, we introduce a new concept of convexity which is generated by a family of endomorphisms of an Abelian group. In Abelian groups equipped with a translation invariant metric, we define the boundedness, the norm, the…

Several results about the union-closed sets conjecture are presented.

In this note we study the maximal perimeter of a convex set in $\mathbb{R}^n$ with respect to various classes of measures. Firstly, we show that for a probability measure $\mu$ on $ \mathbb{R}^n$, satisfying very mild assumptions, there…

The results in the paper are related to the classification problem for invariant subspaces of multiplication operators in several variables. The main results consist of characterizations, in the two dimensional case, of ideals of…

We consider a system of weak* closed sets of finite-dimensional distributions. We show that a corresponding system of random variables can be defined on a probability space with a probability measure determined up to some set of measures,…

We obtain a far-reaching generalization (in several directions) of the theorem of A. Lambert on the existence of the projective tensor product of operator sequence spaces. This result is obtained in the context of spaces, generalizing…

In this article, we investigate some fixed point results satisfying a new generalized $\Delta$-implicit contractive condition in ordered complete multiplicative $\mathbf{G}_\mathcal{M}-$metric space. Also, some new definitions and fixed…

Upper bounds are obtained for the Newtonian capacity of compact sets in $\R^d,\,d\ge 3$ in terms of the perimeter of the $r$-parallel neighbourhood of $K$. For compact, convex sets in $\R^d,\,d\ge 3$ with a $C^2$ boundary the Newtonian…

The class of $\mu$-compact sets can be considered as a natural extension of the class of compact metrizable subsets of locally convex spaces, to which the particular results well known for compact sets can be generalized. This class…

Extending results of Suss and Hadwiger (proved by them for the case of convex bodies and positive ratios), we show that compact (respectively, closed) convex sets in the Euclidean space of dimension n are homothetic provided for any given…

The aim of this paper is to establish a theory of Galerkin approximations to the space of convex and compact subsets of $\R^d$ with favorable properties, both from a theoretical and from a computational perspective. These Galerkin spaces…

In this paper, we give an interesting extension of the partial S-metric space which was introduced [4] to the M_s-metric space. Also, we prove the existence and uniqueness of a fixed point for a self mapping on an Ms-metric space under…

We consider the problem of minimizing convex combinations of the first two eigenvalues of the Dirichlet-Laplacian among open sets of $R^N$ of fixed measure. We show that, by purely elementary arguments, based on the minimality condition, it…

We study random compact subsets of R^3 which can be described as "random Menger sponges". We use those random sets to construct a pair of compact sets A and B in R^3 which are of the same positive measure, such that A can be covered by…

We initiate the study of matrix convexity for operator spaces. We define the notion of compact rectangular matrix convex set, and prove the natural analogs of the Krein-Milman and the bipolar theorems in this context. We deduce a canonical…

We show that, for vector spaces in which distance measurement is performed using a gauge, the existence of best coapproximations in $1$-codimensional closed linear subspaces implies in dimensions $\geq 2$ that the gauge is a norm, and in…

We consider multiple and set-indexed sums of random vectors taking values in Euclidean space of growing dimension. It is shown that, when viewed as finite metric spaces, the sets of values of such sums converge in probability. The limit is…

We study the probability distribution of the area and the number of vertices of random polygons in a convex set $K\subset\mathbb{R}^2$. The novel aspect of our approach is that it yields uniform estimates for all convex sets…

We develop a notion of projections between sets of probability measures using the geometric properties of the 2-Wasserstein space. It is designed for general multivariate probability measures, is computationally efficient to implement, and…

We investigate moment sequences of probability measures on $E\subset\mathbb{R}$ under constraints of certain moments being fixed. This corresponds to studying sections of $n$-th moment spaces, i.e. the spaces of moment sequences of order…