Sharp reach bounds and quantitative localization for opaque sets

Let Q be the unit square and put Λ = √2 + √6/2. We prove two sharp inequalities for finite-length continua in the plane. A set containing two adjacent corners has length, plus its convex-hull distances to the other corners, at least Λ. A corresponding inequality holds for two sets whose hulls meet and which together contain three corners. These bounds extend to finite chains of component hulls, with coefficient one on the total distance between consecutive hulls.

The argument also allows missing projection intervals and applies to every closed opaque set of finite length, regardless of its number of components or its location. After the corners are adjoined, any such barrier shorter than Λ has either four distinct corner components or one component containing an adjacent pair and two further components, one through each remaining corner. In the latter case, we obtain positive lower bounds on the path distances between the three hulls. A tree replacement argument rules out some configurations in the former case. Exact arithmetic verifies the algebraic inequalities for all 105 six-leaf topologies and all 45 five/four-leaf pairs. The other configurations, and the unrestricted lower bound for the square, remain open.