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A closed simple smooth spherical curve dividing the Sphere into two parts of equal areas has at least four Inflection Points.
See also Ball, Baseball Cover
References
Arnold, V. I. Topological Invariants of Plane Curves and Caustics. Providence, RI: Amer. Math. Soc., 1994.
Martinez-Maure, Y. ``A Note on the Tennis Ball Theorem.'' Amer. Math. Monthly 103, 338-340, 1996.