Real-valued Mappings of Spheres

This note concerns subsets A of the unit 2-sphere 5 such that (*) for each continuous real-valued mapping/ of 5 there exists a rotation r of S with all points of r(A) having the same value under/. In 1942, Kakutani [3 ] proved that the set A of end points of an orthonormal set of 3 vectors has property (*). It was observed by de Mira Fernandes [5 ] that the same proof holds in case A is the set of vertices of any equilateral triangle. Yamabe and Yujobo [8] proved a generalization of Kakutani's theorem to ra-space. Their method may be used to prove that the set A of vertices of an isosceles triangle has property (*) (this has been carried out in a Master's thesis of R. D. Johnson [2]). Here we prove that the set A of vertices of any triangle has property (*); the methods differ from both those of Kakutani and those of Yamabe and Yujobo. Dyson [l ] has proved that the set of vertices of a square centered at the origin has property (*); Livesay [4] has extended this to any rectangle centered at the origin. The problem of finding all such sets A having property (*) is unsolved.