Intrinsically quasi-isometric sections in metric spaces

TL;DR

This paper introduces and studies intrinsically quasi-isometric sections in metric spaces, exploring their properties such as large-scale regularity, convexity, and vector space structure, using Cheeger theory and basic mathematical tools.

Contribution

It defines a new class of sections in metric spaces and investigates their properties, including regularity, convexity, and an equivalence relation, advancing large-scale geometric analysis.

Findings

Establishment of large-scale Ahlfors-David regularity for these sections.

Definition of convexity and vector space structure for sections.

Introduction of an equivalence relation among sections.

Abstract

This note is a contribution to large scale geometry. More precisely, we introduce the intrinsically quasi-isometric sections in metric spaces and we investigate their properties: the Ahlfors-David regularity in large scale; following Cheeger theory, it is possible to define suitable sets in order to obtain convexity and being a vector space over $\mathbb{R}$ or $\mathbb{C}$ for these sections; yet, following Cheeger's idea, we give an equivalence relation for this class of sections. Throughout the paper, we use basic mathematical tools.