Vietoris--Rips Contractibility for Dense Subsets of Finite-Dimensional Normed Spaces

Abstract

Let $Y$ be a $\delta$-dense subset of a finite-dimensional normed space $X$, so every point of $X$ is at distance at most $\delta$ from some point of $Y$. We prove that the open Vietoris--Rips complex $\operatorname{VR}_{<}(Y;r)$ is contractible whenever $r>\delta/(1-J_{\mathrm{fin}}(X))$, where $J_{\mathrm{fin}}(X)$ is the normalized finite Jung constant: the supremum of circumradius divided by diameter over finite subsets of positive diameter. We use the canonical isometric embedding of $X$ into a hyperconvex normed space, where $\operatorname{VR}_{<}(Y;r)$ is the nerve of an equal-radius open-ball cover. The finite Jung estimate restricts the cover to a convex tube about $X$ without changing its nerve, while $\delta$-density makes the restricted balls cover the tube. Thus the complex is homotopy equivalent to the tube and hence contractible. Applying this result to lattices gives explicit contractibility bounds. In $(\mathbb{Z}^n,\ell_1)$, gaps in the distance spectrum sharpen the bound: $\operatorname{VR}_{<}(\mathbb{Z}^n,\ell_1;r)$ is contractible above $\max\{n,n(n-1)/2\}$, and $\operatorname{VR}_{\le}(\mathbb{Z}^n,\ell_1;r)$ is contractible at or above the same value. This asymptotically halves the leading term of Zaremsky's previous bound. We also give explicit bounds for $(\mathbb{Z}^n,\ell_p)$ for all $1\le p\le\infty$, recovering the optimal bound $r=1$ for $p=\infty$; for fixed $n$, the bounds converge as $p\to\infty$. For a general locally finite $\delta$-dense subset $Y\subset X$ whose distance spectrum may not be discrete, a separate closed-cover argument proves that $\operatorname{VR}_{\le}(Y;r)$ is contractible whenever $r\ge\delta/(1-J_{\mathrm{fin}}(X))$.