Geometries on Polygons in the unit disc

2023/11/10 by Charalampos Charitos, Charitos, Charalampos, Ioannis Papadoperakis +3
Mathematics · #51F99 #Analytic and geometric function theory #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Holomorphic and Operator Theory #Metric Geometry (math.MG)

paper · pdf · doi:10.48550/arxiv.2311.05968

openalex publication_date 2023/11/10 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

For a family C of properly embedded curves in the 2-dimensional disk \mathbbD² satisfying certain uniqueness properties, we consider convex polygons P⊂ \mathbbD² and define a metric d on P such that (P,d) is a geodesically complete metric space whose geodesics are precisely the curves \ c∩ P\bigm\vert c∈ C\. Moreover, in the special case C consists of all Euclidean lines, it is shown that P with this new metric is not isometric to any convex domain in ℝ ² equipped with its Hilbert metric. We generalize this construction to certain classes of uniquely geodesic metric spaces homeomorphic to ℝ².

Related

• A specific model of Hilbert geometry on the unit disc
• Geometries of convex and finite sets of geodesic spaces
• Intrinsic structure of minimal discs in metric spaces
• Bijections on strictly convex sets and closed convex projective surfaces that preserve complete geodesics
• Metrics in the space of curves