Let $X = \{x_1, \dots, x_n\}$ denote a finite set of $n$ points in the unit square $S$, and let's center $S$ at the origin. Let $F(X) = \sum_{i=1}^n \| x_i \| $ and let $G(X) = \iint_S \min_i \|x - x_i\|~ dA $ be the average distance between a uniformly sampled point in $S$ and its nearest neighbor in $X$. Clearly $F(X)$ increases as $n$ becomes larger and $G(X)$ decreases. For any given configuration $X$, how are these two quantities related (e.g. orders of magnitude)?

Let $X = \{x_1, \dots, x_n\}$ denote a finite set of $n$ points in the unit square $S$, and let's center $S$ at the origin. Let $F(X) = \sum_{i=1}^n \| x_i \| $ and let $G(X) = \iint_S \min_i \|x - x_i\|~ dA $ be the average distance between a uniformly sampled point in $S$ and its nearest neighbor in $X$. Clearly $F(X)$ increases as $n$ becomes larger and $G(X)$ decreases. For any given configuration $X$, how are these two quantities related (e.g. orders of magnitude)? More specifically I'd like to relate these two quantities via inequalities; for example, if $G(X)$ is small, it must mean that the points are somewhat uniformly distributed in $S$, and therefore their average distance from the origin should be somewhere between $1/2$ and $\sqrt{2}/2$. On the other hand if $F(X)$ is small then it means all of the points are packed close to the origin, so $G(X)$ must be fairly large, regardless of $n$.