Separating Measures by Hyperplanes

Abstract

Given red and blue points in $\mathbb{R}^d$, connect every red point to every blue point and weigh each segment by its length under a fixed norm. What fraction of the total weight can always be captured by a single affine hyperplane separating the segments' endpoints? We study this question for finite point configurations in affine general position and, more generally, for pairs of probability measures. For dimension one, we show that choosing the best of three thresholds determined by the means of the two distributions captures at least one third of the total red-blue distance. We also give a matching upper bound. Embedding the line construction to higher dimensions shows that the guarantee of $1/3$ cannot be improved for arbitrary measures. Thus, we pivot to the family of measures whose atoms are in affine general position and whose non-atomic part places zero mass on all hyperplanes. Crucially, we establish that the worst-case guarantee is identical for finite configurations in affine general position and several natural classes of probability measures, thereby unifying the discrete and continuous formulations of the problem. For this family of measures and for higher dimensions, we extend our guarantee of $1/3$ for all norms. Our proofs combine the ham-sandwich theorem with a novel coupling argument and, in the planar case, an integral representation of norms. We also show that the optimal guarantee improves as the dimension grows, giving a lower bound of $1/2-\operatorname{O}\left(1/\sqrt{d}\right)$ as well as an asymptotically matching upper bound of $1/2$ for every $d$. For the Euclidean distance, we refine the lower bounds to $8/(8+5π)$ and $1/2-\operatorname{O}\left(1/d\right)$ and, for $d=2$, the upper bound to $5/11$. We also derive dimension-dependent guarantees for finite point configurations, including perfect separation of $n$ points of each color when $d\geq2n-1$.