The minimum speed for a blocking problem on the half plane

We consider a blocking problem: fire propagates on a half plane with unit speed in all directions. To block it, a barrier can be constructed in real time, at speed σ. We prove that the fire can be entirely blocked by the wall, in finite time, if and only if σ>1. The proof relies on a geometric lemma of independent interest. Namely, let K⊂R2 be a compact, simply connected set with smooth boundary. We define dK(x,y) as the minimum length among all paths connecting x with y and remaining inside K. Then dK attains its maximum at a pair of points both on the boundary of K.