Explicit lower bounds for opaque sets of unit square and unit disc
Abstract
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 [9]. Employing a recent observation by Steinerberger on the possible orientations of straight barriers with length close to Jones’ bound, we improve the bound in [9] 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 [9] can be optimized due to the strong symmetry properties of the disc. Our approach illustrates both the power and the limitations of the method.
Fields of Study & Subjects
Point processes and geometric inequalities
Analytic and geometric function theory
Holomorphic and Operator Theory
Publication details
- Journal / source
- Acta Mathematica Academiae Scientiarum Hungaricae
- Volume
- 178
- Issue
- 1
- Pages
- 289–303
- Publication date
- 1 February 2026
- Language
- EN