Geometric Bounds for Persistence

. In this paper, we offer a new perspective on persistent homology by integrating key concepts from metric geometry. For a given compact subset X of a Banach space Y, we analyze the topological features arising in the family N,pX & Abreve; Yq of nested neighborhoods of X in Y and provide several geometric bounds on their persistence (lifespans). We begin by examining the lifespans of these homology classes in terms of their filling radii in Y, establishing connections between these lifespans and fundamental invariants in metric geometry, such as the Urysohn width. We then derive bounds on these lifespans by considering the P8-principal components of X, also known as Kolmogorov widths. Additionally, we introduce and investigate the concept of extinction time of a metric space X: the critical threshold beyond which no homological features persist in any degree. We propose methods for estimating the Cech and Vietoris-Rips extinction times of X by relating X to its convex hull and to its tight span, respectively.