A Lower Bound on Opaque Sets

It is proved that the total length of any set of countably many rectifiable curves, whose union meets all straight lines that intersect the unit square U, is at least 2.00002. This is the first improvement on the lower bound of 2 by Jones in 1964. A similar bound is proved for all convex sets U other than a triangle

A lower bound on opaque sets

It is proved that the total length of any set of countably many rectifiable curves, whose union meets all straight lines that intersect the unit square U, is at least 2.00002. This is the first improvement on the lower bound of 2 by Jones in 1964. A similar bound is proved for all convex sets U other than a triangle. © Akitoshi Kawamura, Sonoko Moriyama, Yota Otachi, and János Pach

A lower bound on opaque sets

It is proved that the total length of any set of countably many rectifiable curves whose union meets all straight lines that intersect the unit square U is at least 2.00002. This is the first improvement on the lower bound of 2 known since 1964. A similar bound is proved for all convex sets U other than a triangle. (C) 2019 Published by Elsevier B.V

Computational Geometry Column 58

This column is devoted to opaque sets also known as barriers. A set of curves Γ that meet every line which intersects a given convex body B is called an opaque set or barrier for B. Although the shape and length of shortest barriers for simple bodies, such as a unit equilateral triangle or a unit square are seldom disputed, no proof of optimality is known or appears to be even near in sight