Improving Lower Bound on Opaque Set for Equilateral Triangle

TL;DR

This paper improves the lower bound on the length of the shortest opaque set for a unit equilateral triangle, surpassing the classic bound by Jones through novel techniques involving angle restrictions and weighted projection conditions.

Contribution

It introduces a new lower bound of 3/2 plus a tiny positive increment for the barrier length, using innovative methods not previously applied to triangles.

Findings

Established a lower bound of 3/2 + 5×10^{-13} for the barrier length.

Developed new techniques: angle-restricted barriers and weighted sum of projection-cover conditions.

Enhanced understanding of barriers for convex polygons, especially triangles.

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.