A convexity in $\mathbb{R}^2$ with river metric

TL;DR

This paper explores convexity properties in the plane equipped with the river metric, providing a complete characterization of convex sets and analyzing measures of noncompactness within this unique metric space.

Contribution

It introduces a new convexity concept in the river metric space and fully characterizes convex sets, along with studying measures of noncompactness in this context.

Findings

Complete characterization of convex sets in the river metric space

Introduction of $W$-convex structure in $(\,\mathbb{R}^2, d^*)$

Analysis of measures of noncompactness and their moduli

Abstract

In this paper we consider the space $\mathbb{R}^2$ with the river metric $d^*$ and different types of convexity of this space. We define $W$-convex structure in $(\mathbb{R}^2,d^*)$ and we give the complete characterization of the convex sets in this space. We consider some measures of noncompactness and we give the moduli of noncompactness for considered measures on this space.