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arXiv · 2311.05968

Geometries on Polygons in the unit disc

Charalampos Charitos·Ioannis Papadoperakis·Georgios Tsapogas
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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 $\left\{ c\cap P\bigm\vert c\in \mathcal{C}\right\}.$ 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 $\mathbb{R}^{2}.$

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Charalampos Charitos, Ioannis Papadoperakis, Georgios Tsapogas. 2023-11-10. Geometries on Polygons in the unit disc. https://arxiv.org/abs/2311.05968

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