CONTINUOUS FUNCTIONS DEFINED ON SPHERES

This theorem was conjectured some years ago by Kakutani [2]. He proved the case n = 2, generally known as Kakutani's theorem. The object of the present paper is to prove the following analog of Kakutani's theorem. THEOREM. Let S be the surface of a sphere center Z in Euclidean 3-space R3, and let f(x) be a continuous real-valued function defined on S. Then there exist four points X1 , X2 , X3 , X4 on S forming the vertices of a square with center Z, such that