## Curves on a Sphere

Here’s a riddle from Peter Winkler‘s book “Mathematical Mind-Benders” (which is wholeheartedly recommended):

On the surface of a sphere of radius 1 lies a closed curve of length less then $2 \pi$. Prove that it is contained in some hemisphere.

Since there are currently no readers to this probablog, there’s no reason to wait a couple of days before posting hints or solutions, beside procrastination. Well, I guess that’s a good enough reason for me.