Before revision:  We present exact \Re{R}  algorithms for computing the yolk and the egg of a point set, both running in expected time O(n ^{d-1 \log n) .
After revision:  We present exact \Re{R} algorithms for computing the yolk and the egg of a point set, including the centerpoint and the Tukey median, and several other more involved measures of centrality. For d=2, the new algorithms run in O(n ^{d-1 \log n) .