Let $C$ be a continuous curve in the unit square having length $L$. Is there a lower bound on the average distance between the points in the unit square and $C$, as a function of $L$? Is there an asymptotic behavior that's known as $L$ gets large? (other suggestions for tags are welcome)
Here is a rough answer. I think it has to give the right order of magnitude.
$1/L$. If you draw a zigzag curve that goes up and down $L$ times it has length approximately $L$. Each point is distance no more than $1/L$ from the curve.
On the other hand if you consider a neighbourhood of a curve of width $1/(4L)$ on each side, it has area bounded above by approximately $1/2$. This means that 50% of points are at a distance greater than $1/(4L)$ from the curve so the average distance is at least $1/(8L)$.