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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)

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up vote 4 down vote accepted

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 $1/(8L)$.

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Thanks! That's exactly what I needed. –  John Gunnar Carlsson Jan 26 '11 at 8:29

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