Constructing optimal highways

For two points p and q in the plane, a straight line h, called a highway, and a real v > 1, we define the travel time (also known as the city distance) from p and q to be the time needed to traverse a quickest path from p to q, where the distance is measured with speed v on h and with speed 1 in the underlying metric elsewhere. Given a set S of n points in the plane and a highway speed v, we consider the problem of finding a highway that minimizes the maximum travel time over all pairs of points in S. If the orientation of the highway is fixed, the optimal highway can be computed in linear time, both for the L1- and the Euclidean metric as the underlying metric. If arbitrary orientations are allowed, then the optimal highway can be computed in O(n2 log n) time. We also consider the problem of computing an optimal pair of highways, one being horizontal, one vertical.