Improving Lower Bound on Opaque Set for Equilateral Triangle

Abstract: An opaque set (or a barrier) for $U \subseteq \mathbb{R}^2$ is a set $B$ of finite-length curves such that any line intersecting $U$ also intersects $B$. In this paper, we consider the lower bound for the shortest barrier when $U$ is the unit equilateral triangle. The known best lower bound for triangles is the classic one by Jones [Jones,1964], which exhibits that the length of the shortest barrier for any convex polygon is at least the half of its perimeter. That is, for the unit equilateral triangle, it must be at least $3/2$. Very recently, this lower bounds are improved for convex $k$-gons for any $k\geq 4$ [Kawamura et al. 2014], but the case of triangles still lack the bound better than Jones' one. The main result of this paper is to fill this missing piece: We give the lower bound of $3/2 + 5 \cdot 10^{-13}$ for the unit-size equilateral triangle. The proof is based on two new ideas, angle-restricted barriers and a weighted sum of projection-cover conditions, which may be of independently interest.