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We show that the Vietoris-Rips complex $\mathcal R(n,r)$ built over $n$points sampled at random from a uniformly positive probability measure on aconvex body $K\subseteq \mathbb R^d$ is a.a.s. contractible when $r \geq c\left(\frac{\ln n}{n}\right)^{1/d}$ for a certain constant that depends on $K$and the probability measure used. This answers a question of Kahle [DiscreteComput. Geom. 45 (2011), 553-573]. We also extend the proof to show that if $K$is a compact, smooth $d$-manifold with boundary - but not necessarily convex -then $\mathcal R(n,r)$ is a.a.s. homotopy equivalent to $K$ when $c_1\left(\frac{\ln n}{n}\right)^{1/d} \leq r \leq c_2$ for constants $c_1=c_1(K),c_2=c_2(K)$. Our proofs expose a connection with the game of cops and robbers.

On the contractibility of random Vietoris-Rips complexes