Two planar facility location problems with high-speed corridors and continuous demand

The planar minisam (‘median’) and minimax (‘center’) facility location problems are examined under the assumptions that: demand is continuously and uniformly distributed: the L₁ (right-angle) metric is in use; and the planar region is traversed by a high-speed corridor (highway) running parallel to one of the directions of travel. For the minisum problem in a rectangular region, it is shown that, for any combination of problem parameters, there are only two candidate points in the region for the optimum location of the facility. This is also shown to be true for any convex and symmetric planar region. For the minimax problem in a rectangular region, there are only three candidate points for the optimal location. Some extensions and conjectures for the minisum problem involving more than one highway are also discussed.