Their first general formulation is as follows: for any suitably restricted real function f \colon S^n \times X \to \mathbb{R}, there is a point of the -sphere such that the surface , dividing into and , simultaneously bisects the outer measure of .
Their second formulation is as follows: for any measurable functions over that are linearly independent over any subset of of positive measure, there is a linear combination such that the surface , dividing into and , simultaneously bisects the outer measure of .