2025/01/02 by Steinerberger, Stefan
#FOS: Mathematics #Metric Geometry (math.MG)
paper · doi:10.48550/arxiv.2501.01004
Let Ω⊂ ℝ2 be a bounded, convex set. A set O ⊂ ℝ2 is an opaque set (for Ω) if every line that intersects Ω also intersects O. What is the minimal possible length L of an opaque set? The best lower bound L ≥ |∂ Ω|/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 - |∂ Ω|/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 ∂ Ω in a precise sense.