A problem related to Bárány-Grünbaum conjecture

Abstract

In this paper we prove that for any absolute continuous Borel probability measure μ on the sphere S2 and any t ∈ [0; 1/4 ] there exist four great semi-circles l1,..., l4 emanating from a point x 2 S2 that partition sphere S2 into four angular sectors σ1,..., σ4, counter clockwise oriented, such that μ(σ1) = μ(σ4) = t; μ(σ2) = μ(σ3) = 1/4 - t, and area(σ1) = area(σ4); area(σ2) = area(σ3).