Fast Algorithms for Geometric Consensuses

Let $P$ be a set of $n$ points in $\Re^d$ in general position. A median hyperplane (roughly) splits the point set $P$ in half. The yolk of $P$ is the ball of smallest radius intersecting all median hyperplanes of $P$. The egg of $P$ is the ball of smallest radius intersecting all hyperplanes which contain exactly $d$ points of $P$. We present exact algorithms for computing the yolk and the egg of a point set, both running in expected time $O(n^{d-1} \log n)$. The running time of the new algorithm is a polynomial time improvement over existing algorithms. We also present algorithms for several related problems, such as computing the Tukey and center balls of a point set, among others.