3

This problem is giving me the hardest time:

Alex is lost in a forest. The forest area has a convex shape whose area is $P$. Prove that Alex can choose a path not longer than $\sqrt{2 \pi P}$ such that he will exit the forest for sure.

This is from a set of problems for competition training.

VividD
  • 15,966

1 Answers1

1

If Alex traverses a path $\Gamma$ without ever leaving the interior of the forest, then the entire path is contained in the forest; and since the forest is convex, its area must then be greater than the area inside the convex hull of $\Gamma$. So you need to find a curve of length $\sqrt{2\pi P}$ (not necessarily closed) whose convex hull contains area at least $P$. One possibility is a semicircle with radius $r$, which contains area $\frac{1}{2}\pi r^2 = P$ provided that $r = \sqrt{2 P / \pi}$. In that case, the length of the curve is exactly $\pi r = \sqrt{2 \pi P}$, so you are done.

mjqxxxx
  • 41,358
  • Thanks - however, my understanding is that P is not known to Alex in advance. – VividD Sep 22 '14 at 21:43
  • But probably I am wrong. I'll have to check with the problem author. So I will mark your answer as accepted, so that you and other people do not waste time... – VividD Sep 22 '14 at 21:53
  • That doesn't seem consistent with how the question is worded… it is written as if Alex is to "choose a path not longer than $\sqrt{2\pi P}$", i.e., he knows how long of a path he must choose. I don't think there is a strategy for Alex such that he can guarantee he will leave the forest within a distance $\sqrt{2\pi P}$, where $P$ is the (unknown) area of the forest, but I would be interested to see such a solution. – mjqxxxx Sep 22 '14 at 21:56
  • Meanwhile, it would be interesting to compare semicircle to other possible paths - semisquare or similar. – VividD Sep 22 '14 at 21:56
  • I think your interpretation is right. – VividD Sep 22 '14 at 22:03