The Weighted Euclidean 1-Center Problem

We present an O(n(log n)³(log log n)²) algorithm for the problem of finding a point (x, y) in the plane that minimizes the maximal weighted distance to a point in a set of n given points. The algorithm can be extended to higher dimensional spaces. For any fixed dimension our bound is o(n^1+ϵ) for any ϵ > 0.