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) . The running timeof 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 .
After 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) expected time, which is optimal, and for higher constant d>2, the expected time bound is within one logarithmic factor of O(n^{d-1 .