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 $\Bbb{R}^{3}$. It was conjectured that for every triple of rationalnumbers $(q_{1},q_{2},q_{3})$ with the property $q_{1}+q_{2}+q_{3}=\tfrac{1}{2}$,there exist three planes in $\Bbb{R}^{3}$ 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.