We are given m pursuers and one evader. Each pursuer and the evader has an associated starting point in the plane, a maximum speed, and a start time. We also have a set of line segment obstacles with a total of n endpoints. Our task is to find those points in the plane, called the evader?s region, that the evader can reach via evasive paths. A path is evasive if the evader can traverse the path from its starting point without encountering a pursuer along the way. The evader and the pursuers must obey their start time and speed constraints, and cannot go through obstacles. The partition of the plane into the evader?s region and the remaining pursuers? region is called the pursuit-evasion Voronoi diagram.

We study pursuit-evasion Voronoi diagrams for the \ell_1 metric. We show that the complexity of the diagram is O((n +m)^2(mn+m)) and that it can be calculated in polynomial time.