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 d_K (x, y) as the minimum length among all paths connecting x with y and remaining inside K. Then d_K attains its maximum at a pair of points (over(x, ̄), over(y, ̄)) both on the boundary of K.