Related papers: A Short Study of Alexandroff Spaces

The first section of this modest survey reviews some basic notions and describes some families of examples, and the second section briefly indicates some general aspects of analysis on metric spaces. The remaining three sections are…

This paper aims to give an elementary proof for Toponogov's theorem in Alexandrov geometry with lower curvature bound. The idea of the proof comes from the fact that, in Riemannian geometry, sectional curvature can be embodied in the second…

We show a new, elementary and geometric proof of the classical Alexandrov theorem about the second order differentiability of convex functions. We also show new proofs of recent results about Lusin approximation of convex functions and…

Bielavsky introduced and investigated the class of symmetric symplectic spaces, that is, symmetric spaces endowed with a symplectic form invariant with respect to symmetries. Since the theory of symmetric spaces has generalizations, we ask…

We give a simplified proof of the generalized Kirszbraun theorem for Alexandrov spaces, which is due to Lang and Schroeder. We also discuss related questions, both solved and open.

We obtain new topological information about the local structure of collapsing under a lower sectional curvature bound. As an application we prove a new sphere theorem and obtain a partial result towards the conjecture that not every…

The present paper is a detailed and almost self-contained introduction to Kazarnovskii mixed pseudovolume. It provides new formulas and new results about Alexandroff-Fenchel type inequalities for Kazarnovskii mixed pseudovolume.

We give a short introduction to the theory of modular metric spaces. This is a corrected version of the paper [1], which had some errors. We are grateful to V. V. Chistyakov for bringing these to our attention.

Here we have studied the idea of semi Lamda*-closed sets and investigate some of their properties in spaces considered by A. D. Alexandroff [1]. We have introduced some new separation axioms namely semi-Tw/4, , semi-T3w/8, semi-T5w/8, and…

In this paper we focus on minimal Besicovitch arrangements to highlight some of their properties. An appropriate probability space enables us to find again in an elegant way some straightforward equalities associated with these…

Here I give a description of Alexandrov 4-point comparison via quadratic forms and then propose a natural 5-point condition which might have some future. Consider this note as a letter from me --- do not take it seriously.

We provide new examples of integrable rational maps in four dimensions with two rational invariants, which have unexpected geometric properties, as for example orbits confined to non algebraic varieties, and fall outside classes studied by…

This paper provides a systematic exposition of Lyapunov stability for compact sets in locally compact metric spaces. We explore foundational concepts, including neighborhoods of compact sets, invariant sets, and the properties of dynamical…

In this paper, we define the fractional Orlicz-Sobolev spaces, and we prove some important results of these spaces. The main result is to show the continuous and compact embedding for these spaces. As an application, we prove the existence…

The notes provide a short introduction to de Branges--Rovnyak spaces. They cover some basic facts and are intended to give the reader a taste of the theory, providing sufficient motivation to make it interesting.

We study submetries between Alexandrov spaces and show how some of the usual features of Riemannian submersions fail due to the lack of smoothness.

We extend basic results in $3$-manifold topology to general three-dimensional Alexandrov spaces (or Alexandrov $3$-spaces for short), providing a unified framework for manifold and non-manifold spaces. We generalize the connected sum to…

We use pointwise Kan extensions to generate new subcategories out of old ones. We investigate the properties of these newly produced categories and give sufficient conditions for their cartesian closedness to hold. Our methods are of…

The aim of these lecture notes is, after having quickly described various compactifications of the Teichm\"{u}ller space of a compact connected oriented surface minus finitely many points, to give a construction, by the equivariant Gromov…

It is proved the generalization of Toponogov theorem about the length of the curve in two-dimensional Riemannian manifolds in the case of two-dimensional Alexandrov spaces.