Geometries on Polygons in the unit disc

Abstract

For a family $\mathcal{C}$ of properly embedded curves in the 2-dimensional disk ${\mathbb{D}}^2$ satisfying certain uniqueness properties, we consider convex polygons $P\subset {\mathbb{D}}^2$ and define a metric $d$ on $P$ such that $(P,d)$ is a geodesically complete metric space whose geodesics are precisely the curves $\left\{c\cap P|c\in \mathcal{C}\right\}.$ Moreover, in the special case $\mathcal{C}$ consists of all Euclidean lines, it is shown that $P$ with this new metric is not isometric to any convex domain in ${\mathbb{R}}^2$ equipped with its Hilbert metric. We generalize this construction to certain classes of uniquely geodesic metric spaces homeomorphic to ${\mathbb{R}}^2.$

To appear in Rocky Mountain J. Math