## Curves on a Sphere – a Solution and a Hint

First, the solution. This is not one of the two solutions appearing in the book, which are actually nicer.

The curve separates the surface of the sphere into 2 regions. Paint the smaller one black and the other white. Pick (one of) the hemisphere which maximizes the black area in it (there’s a global maximum because of continuity and compactness, but a local maximum also suffices). We claim that the entire black area (hence the curve) is contained in this hemisphere.

Why is that so? Consider the circle circumventing that hemisphere. If it does not intersect the curve, we’re done. Suppose it does intersect the curve. Then all the points of intersection can’t be contained in just one half of the circle, for then a slight rotation of the hemisphere would increase the black area in it. Therefore, there are several points of intersection, such that the shortest distance between any two consecutive points in along the circle. But this sums up to $2 \pi$, contradicting our assumption.

So, what’s with the hint? Isn’t it supposed to come before the solution? Not if the hint is for another solution!

And the hint is this:

Why would this riddle appear in a probablog?