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