Explicit lower bounds for opaque sets of unit square and unit disc

Abstract

Explicit lower bounds for the length of the shortest opaque set for the unit disc and the unit square in the Euclidean plane are derived. The results are based on an explicit application of the general method of Kawamura, Moriyama, Otachi and Pach. Employing a recent observation by Steinerberger on the possible orientations of straight barriers with length close to Jones' bound, we improve the bound for the unit square by more than a factor $3$. The bound for barriers of the unit disc is new and based on the idea that the free parameters in the general method from can be optimized due to the strong symmetry properties of the disc. Our approach illustrates both the power and the limitations of the method.