Weighted partitions of sphere measures by hyperplanes

Abstract

Abstract. In this paper we use the CS / TM scheme stated in [1] for the V.V. Makeev equipartition problem [7] to prove the existence of new weighted partitions. For the first time computations of the associated equivariant problem are done by codes written in the package Mathematica 5.0 [2]. Statement of the main resultLet H 1 , H 2 and H 3 be planes in R 3 through the origin. They are in the fan position if H 1 ∩ H 2 = H 1 ∩ H 3 = H 2 ∩ H 3 . Planes in the fan position cut the sphere S 2 in six parts σ 1 , .., σ 6 which can be naturally oriented up to a cyclic permutation. A measure µ is the proper measure µ if µ([a, b]) = 0 for any circular arc [a, b] ⊂ S 2 , and µ(U ) > 0 for each nonempty open set U ⊂ S 2 . We prove the existence of the following measure partitions. Theorem 1. Let µ be a proper Borel probability measure on the sphere S 2 . Then there are three planes in the fan position such that the ratio of measure µ in angular sectors cut by planes is (A) (1, 1, 2, 1, 1, 2) (C) (1,1,3,1,1,3) This result is a generalization of the Makeev result [7].