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?
1. **Understanding the Problem:**
   Farmer John ties his goat to \( k \) ropes of varying lengths, each centered at points \( Q_1, Q_2, \ldots, Q_k \). The goat can reach any point \( R \) such that \( \ell_j \geq RQ_j \) for all \( j \in \{1, 2, \ldots, k\} \). Given a finite set of points \( P_1, P_2, \ldots, P_n \) where grass is planted, we need to determine the largest possible lower bound on the area of the region that the goat can reach.

2. **Convex Hull Concept:**
   The convex hull of a set of points is the smallest convex polygon that contains all the points. If the goat can reach all the points \( P_1, P_2, \ldots, P_n \), it must be able to reach any point inside the convex hull of these points. This is because the intersection of convex sets is convex, and the region the goat can reach must be convex.

3. **Case Analysis:**
   - **Case 1: All Points are Collinear**
     If all points \( P_1, P_2, \ldots, P_n \) are collinear, they lie on a line segment, say \(\overline{P_1P_n}\). In this case, the convex hull is simply the line segment \(\overline{P_1P_n}\). The area of this convex hull is zero because it is a one-dimensional object.

   - **Case 2: Points Form a Nonzero Area Convex Hull**
     If the points form a convex hull with nonzero area, let the convex hull be \( P_1P_2 \ldots P_k \). We can construct circles \(\omega_i\) passing through \( P_i \) and \( P_{i+1} \) (with indices taken modulo \( k \)). The centers of these circles are outside the convex hull but on the same side of \( P_iP_{i+1} \) as the rest of the points. As the radii of these circles tend to infinity, the area of their intersection tends to the area of the convex hull.

4. **Conclusion:**
   Since the intersection of convex sets is convex, the region in which the goat can freely move must be convex. Therefore, the goat can reach all of the points \( P_1, P_2, \ldots, P_n \) if and only if it can reach any point inside the convex hull of these points. This proves that the largest possible lower bound on the area of the region that the goat can reach is the area of the convex hull of the points.

\[
\boxed{\text{Area of the convex hull of } P_1, P_2, \ldots, P_n}
\]