A Stability Version of the Jones Opaque Set Inequality
A stability result for the Jones (1962) lower bound L >= |boundary|/2 on the length of an opaque set (a set meeting every line through a bounded convex planar domain). Encoding a family of line segments as an angular-orientation measure on the circle (scaled Dirac masses at each segment's angle and its opposite), the theorem bounds the homogeneous negative Sobolev distance in H^{-2}(T) between the opaque set's measure and the boundary's measure by a constant times (L - |boundary|/2)^{3/4}. Consequently, when the Jones bound is nearly attained the opaque set's segments must be oriented almost exactly like the boundary; a direct elementary proof is given for the unit square.
A STABILITY VERSION OF THE JONES OPAQUE SET INEQUALITY STEFAN STEINERBERGER Abstract. Let Ω⊂R2 be a
An opaque set for a bounded convex domain O in R^2 is a set meeting every line that intersects the domain; the classical Jones (1962) bound says its length L is at least |boundary|/2, and improving t…