They sought it with thimbles, they sought it with care; They pursued it with forks and hope; They threatened its life with a railway-share; They charmed it with smiles and soap. – The Hunting of the Snark,

A set of k balls B1,..., Bk in a Euclidean space is said to cover a collection of lines if every line intersects some ball. We consider the k-center problem for lines in high dimensional space: Given a set of n lines L = {ℓ1,..., ℓn} in Rd, find k balls of minimum radius which cover L. We present a 2-approximation algorithm for the cases k = 2, 3 of this problem, having running time quasi-linear in the number of lines and the dimension of the ambient space. Our result for 3-clustering is strongly based on a new result in discrete geometry which may be of independent interest: a Helly-type theorem for collections of axis-parallel “crosses ” in the plane. The family of crosses does not have finite Helly number in the usual sense. Our Helly theorem is of a new type: it depends on ε-contracting the sets. In statistical practice, data is often incompletely specified; we consider lines as the most elementary case of incompletely specified data points. Clustering of data is a key primitive in nonparametric statistics. Our results provide a way of performing this primitive on incomplete data, as well as imputing the missing values.