Farmer John ties his goat to a number of ropes of varying lengths in the Euclidean plane. If he ties the goat to $k$ ropes centered at $Q_1$, $Q_2$, ... $Q_k$ with lengths $\ell_1$, $\ell_2$, ... $\ell_k$ (respectively), the goat can reach any point $R$ such that $\ell_j \geq RQ_j$ for all $j \in \{1,2,3, \ldots k\}$.

Suppose that Farmer John has planted grass at a finite set of points $P_1$, $P_2$, ... $P_n$, and sets the ropes such that the goat can reach all of these points. What is, in terms of the points, the largest possible lower bound on the area of the region that the goat can reach?