Optimal Algorithms for Geometric Centers and Depth

We develop a general randomized technique for solving implicit linear programming problems, where the collection of constraints are defined implicitly by an underlying ground set elements. In many cases, structure can be used to obtain faster program solvers. apply this near-optimal algorithms variety fundamental problems in geometry. For given point $P$ size $n$ $\mathbb{R}^d$, we computing geometric centers set, including centerpoint and Tukey median, several other more involved measures centrality. $d=2$, new run $O(n\log n)$ expected time, which is optimal, higher constant $d>2$, time bound within one logarithmic factor $O(n^{d-1})$, also likely near optimal some problems.