Suppose you get in your car and take a drive on the sphere of radius R, so that when you return to your starting point the odometer indicates you've traveled less than 2πR. Does your path, γ, have to lie in some hemisphere?
This question was presented to us by Dr. Robert Foote of Wabash College. Previous authors chose two points, A and B, on γ such that these points divided γ into two arcs of equal length. Then they took the midpoint of the great circle arc joining A and B to be the North Pole and showed that the curve must be contained in the Northern Hemisphere. This type of proof not only answers the existence question, but also yields a specific hemisphere that contains your path.
We, however, thought the problem lent itself nicely to integral geometry, which required us to consider the space whose points are hemispheres. This led to a different existence proof and to a solution of the more general question: can you describe and measure the set of all hemispheres that contain γ?
An outline of the remainder of this paper follows. In Section 2 we introduce terminology and definitions. The existence of at least one hemisphere containing γ is proved using the ideas of integral geometry in Section 3. Classifying sets of such hemispheres for a single arc, a geodesic triangle, and a geodesic quadrilateral is accomplished in Section 4. Section 5 contains a discussion of convexity on the sphere and how it relates to our question. Our main theorem is stated and proved in Section 6.
Mary Kate Boggiano and Mark Desantis. The Set of Hemispheres Containing a Closed Curve on the Sphere. Technical paper (TR-98-01). Math and Computer Science Technical Report Series. Richmond, Virginia: Department of Mathematics and Computer Science, University of Richmond, February, 1998.