Improving the lower bound on opaque sets for equilateral triangle

An opaque set (or a barrier) for U⊆R2 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 on the shortest barrier when U is the unit-size equilateral triangle. The best known lower bound is 3/2, which comes from the classical fact that the length of the shortest barrier for any convex shape is at least the half of its perimeter. While such a general lower bound is slightly improved very recently, its applicability range does not cover the case of triangles. The main result of this paper is to find out this missing piece in part: We give the lower bound of 3/2+5⋅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 independent interest.