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} We develop a general randomized technique for solving "implic it" linear programming problems, where the collection of constraints are defined implicitly by an underlying ground set of elements. In many cases, the structure of the implicitly defined constraints can be exploited in order to obtain efficient linear program solvers. We apply this technique to obtain near-optimal algorithms for a variety of fundamental problems in geometry. For a given point set P of size n in \Re^d, we develop algorithms for computing the yolk and the egg of a point set, both running in expected time O(n ^{d-1 \log n) .