Reduced polygons in the hyperbolic plane

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 $\hat{I}”(C)$ of $C$. A convex body $R\subset {\mathbb{H}}^d$ is said to be reduced if $\hat{I}”(Z)<\hat{I}”(R)$ 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.

9 pages, 2 figures