Getting out of a convex forest 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.

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.