Guarding a Line Segment

A line segment on the plane is guarded by a defender. An invader wants to pass through the line segment but he has to keep the distance from the defender no less than a given constant ρ > 0. The defender can move on the whole plane with maximal speed 1. The invader can move on the whole plane with maximal speed θ, greater than 1. No further kinematic or dynamic constraints are imposed on the defender and the invader motions. The maximal length of the line segment which can be guarded by the defender is established in this paper.