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 X of a Banach space Y Y , we analyze the topological features arising in the family N ∙ ( X ⊂ Y ) N_• (X⊂ Y) of nested neighborhoods of X X in Y 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 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 ℓ ∞ ^∞ -principal components of X X , also known as Kolmogorov widths. Additionally, we introduce and investigate the concept of extinction time of a metric space X X : the critical threshold beyond which no homological features persist in any degree. We propose methods for estimating the Čech and Vietoris–Rips extinction times of X X by relating X X to its convex hull and to its tight span, respectively.