On the Yao-Yao partition theorem

Abstract.

The Yao-Yao partition theorem states that for any probability measure μ on ${\mathbb{R}}^n$ having a density which is continuous and bounded away from 0, it is possible to partition ${\mathbb{R}}^n$ into 2ⁿ regions of equal measure for μ in such a way that every affine hyperplane of ${\mathbb{R}}^n$ avoids at least one of the regions. We give a constructive proof of this result and extend it to slightly more general measures.