A Stability Version of the Jones Opaque Set Inequality

Let $\Omega \subset \mathbb{R}^2$ be a bounded, convex set. A set $O \subset \mathbb{R}^2$ is an opaque set (for $\Omega$) if every line that intersects $\Omega$ also intersects $O$. What is the minimal possible length $L$ of an opaque set? The best lower bound $L \geq |\partial \Omega|/2$ is due to Jones (1962). It has been remarkably difficult to improve this bound, even in special cases where it is presumably very far from optimal. We prove a stability version: if $L - |\partial \Omega|/2$ is small, then any corresponding opaque set $O$ has to be made up of curves whose tangents behave very much like the tangents of the boundary $\partial \Omega$ in a precise sense.