Well-Separation and Hyperplane Transversals in High Dimensions

A family of $k$ point sets in $d$ dimensions is well-separated if the convex hulls of any two disjoint subfamilies can be separated by a hyperplane. Well-separation is a strong assumption that allows us to conclude that certain kinds of generalized ham-sandwich cuts for the point sets exist. But how hard is it to check if a given family of high-dimensional point sets has this property? Starting from this question, we study several algorithmic aspects of the existence of transversals and separations in high-dimensions. First, we give an explicit proof that $k$ point sets are well-separated if and only if their convex hulls admit no $(k - 2)$-transversal, i.e., if there exists no $(k - 2)$-dimensional flat that intersects the convex hulls of all $k$ sets. It follows that the task of checking well-separation lies in the complexity class coNP. Next, we show that it is NP-hard to decide whether there is a hyperplane-transversal (that is, a $(d - 1)$-transversal) of a family of $d + 1$ line segments in $\mathbb{R}^d$, where $d$ is part of the input. As a consequence, it follows that the general problem of testing well-separation is coNP-complete. Furthermore, we show that finding a hyperplane that maximizes the number of intersected sets is NP-hard, but allows for an $\Omega\left(\frac{\log k}{k \log \log k}\right)$-approximation algorithm that is polynomial in $d$ and $k$, when each set consists of a single point. When all point sets are finite, we show that checking whether there exists a $(k - 2)$-transversal is in fact strongly NP-complete.