Tennis ball theorem

In geometry, the tennis ball theorem states that any smooth curve on the surface of a sphere that divides the sphere into two equal-area subsets without touching or crossing itself must have at least four inflection points, points at which the curve does not consistently bend to only one side of its tangent line.

[1] The tennis ball theorem was first published under this name by Vladimir Arnold in 1994,[2][3] and is often attributed to Arnold, but a closely related result appears earlier in a 1968 paper by Beniamino Segre, and the tennis ball theorem itself is a special case of a theorem in a 1977 paper by Joel L.

[4][5] The name of the theorem comes from the standard shape of a tennis ball, whose seam forms a curve that meets the conditions of the theorem; the same kind of curve is also used for the seams on baseballs.

[1] The tennis ball theorem can be generalized to any curve that is not contained in a closed hemisphere.

must contain points of the curve that belong to both of the hemispheres separated by this great circle.

curve that partitions the sphere into two equal-area components has at least four inflection points in this sense.

However, the condition that the curve divide the sphere's surface area equally is a necessary part of the theorem.

[1] One proof of the tennis ball theorem uses the curve-shortening flow, a process for continuously moving the points of the curve towards their local centers of curvature.

Additionally, as the curve flows, its number of inflection points never increases.

Because curve-shortening does not change any other great circle, the first term in this series is zero, and combining this with a theorem of Sturm on the number of zeros of Fourier series shows that, as the curve nears this great circle, it has at least four inflection points.

[8][9] A generalization of the tennis ball theorem applies to any simple smooth curve on the sphere that is not contained in a closed hemisphere.

As in the original tennis ball theorem, such curves must have at least four inflection points.

It is also analogous to a theorem of August Ferdinand Möbius that every non-contractible smooth curve in the projective plane has at least three inflection points.