  Let $X_1,X_2, \ldots $ be independent identically distributed random points
in a convex polytopal domain $A \subset \mathbb{R}^d$. Define the largest
nearest neighbour link $L_n$ to be the smallest $r$ such that every point of
$\mathcal X_n:=\{X_1,\ldots,X_n\}$ has another such point within distance $r$.
We obtain a strong law of large numbers for $L_n$ in the large-$n$ limit. A
related threshold, the connectivity threshold $M_n$, is the smallest $r$ such
that the random geometric graph $G(\mathcal X_n, r)$ is connected. We show that
as $n \to \infty$, almost surely $nL_n^d/\log n$ tends to a limit that depends
on the geometry of $A$, and $nM_n^d/\log n$ tends to the same limit.
