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arXiv · 2401.07831

Reduced polygons in the hyperbolic plane

Marek Lassak
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Abstract

For a hyperplane $H$ supporting a convex body $C$ in the hyperbolic space $\mathbb{H}^d$ we define the width of $C$ determined by $H$ as the distance between $H$ and a most distant ultraparallel hyperplane supporting $C$. The minimum width of $C$ over all supporting $H$ is called the thickness $Δ(C)$ of $C$. A convex body $R \subset \mathbb{H}^d$ is said to be reduced if $Δ(Z) < Δ(R)$ for every convex body $Z$ properly contained in $R$. We describe a class of reduced polygons in $\mathbb{H}^2$ and present some properties of them. In particular, we estimate their diameters in terms of their thicknesses.

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Marek Lassak. 2024-06-06. Reduced polygons in the hyperbolic plane. https://arxiv.org/abs/2401.07831

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