Suppose that $S_1,\dots,S_n$ is a collection of disjoint shapes in the plane, and let $\mathcal{X}$ denote the set of all $n$-tuples of points $\lbrace x_1,\dots,x_n\rbrace$ such that $x_i\in S_i$ for each $i$. For any such tuple $X = \lbrace x_1,\dots,x_n\rbrace$, let $F(X)$ be defined as

$F(X) = \max_{i} \min_{j\neq i} \|x_i - x_j\| $,

i.e. the maximum nearest-neighbor distance between the points $x_i$. Is there a lower bound for the quantity $\max_{X \in \mathcal{X}} F(X)$? Put another way, I'd like to find the tightest possible value of $r$ in the following statement:

"Suppose that $S_1,\dots,S_n$ is a collection of disjoint shapes in the plane. Then there exists an $n$-tuple of points $\lbrace x_1,\dots,x_n\rbrace$ such that $x_i \in S_i$ for each $i$ and an index $i^*$ such that $ \| x_{i^*} - x_j \| \geq r $ for all indices $j\neq i^*$."

I am conjecturing that the answer is inversely proportional to $\sqrt{n}$ and proportional to the square root of the total area of the regions.