An effective criterion for the existence of a mass partition

Abstract

Let $\mu$ be a proper Borel probability measure on the sphere $S^2$in ${\mathbb{R}}^3$. It was conjectured that for every triple of rationalnumbers $(q_1,q_2,q_3)$ with the property $q_1+q_2+q_3=\frac{1}{2}$,there exist three planes in ${\mathbb{R}}^3$ intersecting along thecommon line through the origin such that the six angular sectors on thesphere determined by those planes have respectively $q_1$, $q_2$, $q_3$,$q_1$, $q_2$, $q_3$ amount of the measure $\mu$. In this paper wegive an exact and explicitly realized algorithm which, for every triple$(q_1,q_2,q_3)$ of the form $q_2=q_3$, establishes whether thereexists a configuration of three planes splitting the measure in the requiredproportion.