GEOMETRIES ON POLYGONS IN THE UNIT DISK
Abstract
For a family π of properly embedded curves in the 2-dimensional disk π»2 satisfying certain uniqueness properties, we consider convex polygons Pβπ»2 and define a metric d on P such that (P,d) is a geodesically complete metric space whose geodesics are precisely the curves {cβ©Pβ£cβπ}. Moreover, in the special case π consists of all Euclidean lines, it is shown that P with this new metric is not isometric to any convex domain in β2 equipped with its Hilbert metric. We generalize this construction to certain classes of uniquely geodesic metric spaces homeomorphic to β2.
Fields of Study & Subjects
Mathematics and Applications
Geometric and Algebraic Topology
Geometric Analysis and Curvature Flows
Publication details
- Journal / source
- Rocky Mountain Journal of Mathematics
- Volume
- 56
- Issue
- 1
- Pages
- 75β90
- Publication date
- 1 February 2026