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 \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 .
 \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 geometric centers 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 \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 .
D
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
A
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
R
the yolk and the egg
geometric centers
R
both running in expected time
including the centerpoint and the Tukey median, and several other more involved measures of centrality. For d=2, the new algorithms run in
D
^{d-1
R
. 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
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