Intrinsically quasi-isometric sections in metric spaces

Abstract

This note contributes to the study of large-scale geometry. Specifically, we introduce the concept of intrinsically quasi-isometric sections in metric spaces and investigate their properties. In particular, we examine their Ahlfors-David regularity at large scales. Building on Cheeger's theory, we define appropriate sets that enable the determination of convexity and establish whether these sections form a vector space over $\mathbb{R}$ or $\mathbb{C}$. Furthermore, inspired by Cheeger's approach, we propose an equivalence relation for this class of sections. Throughout the paper, we employ fundamental mathematical tools.