Diameter, width and thickness in the hyperbolic plane

Abstract

AbstractIn hyperbolic geometry there are several concepts to measure the breadth or width of a convex set. In the first part of the paper we collect them and compare their properties. Than we introduce a new concept to measure the width and thickness of a convex body. Correspondingly, we define three classes of bodies, bodies of constant with, bodies of constant diameter and bodies having the constant shadow property, respectively. We prove that the property of constant diameter follows to the fulfilment of constant shadow property, and both of them are stronger as the property of constant width. In the last part of this paper, we introduce the thickness of a constant body and prove a variant of Blaschke’s theorem on the larger circle inscribed to a plane-convex body of given thickness and diameter.