Planar geometric location problems and maintaining the width of a planar set

We present an O(n² log³ n) algorithm for the two-center problem, in which we are given a set 5 of n points in the plane and wish to find two closed discs whose union contains S so that the larger of the two radii is as small as possible. We also give an O(n² log⁵ n) algorithm for solving the two-line center problem, where we want to find two strips that cover S whose maximum width is as small as possible. To obtain the second result, we need as a subroutine an algorithm for determining whether the width of a given set S of points in the plane ever becomes less than or equal to a given parameter W > 0, as we perform n insertions and deletions on S, and the sequence of these operations is known in advance. We present an O(n log³ n) algorithm for solving this problem.