How to get close to the median shape

In this paper, we study the problem of Li-fitting a shape to a set of n points in ℝ^d (where d is a fixed constant), where the target is to minimize the sum of distances of the points to the shape, or alternatively the sum of squared distances. We present a general technique for computing a (1+ε)-approximation for such a problem, with running time O(n + poly(log n, 1/ε)), where poly(log n, 1/ε) is a polynomial of constant degree of log n and 1/ε (the power of the polynomial is a function of d). This is a linear time algorithm for a fixed ε > 0, and is the first subquadratic algorithm for this problem. Applications of the algorithm include best fitting either a circle, a sphere or a cylinder to a set of points when minimizing the sum of distances (or squared distances) to the respective shape.