Support function and hyperbolic plane

Minkowskis theorem _C(cosh d(o, ) – kS) ds = 0 in the hyperbolic plane (Kleins model) for smoothly bounded horocyclic convex bodies K with outer unit normal vector u and curvature |k| 1 of C K with arclength s where S d(o, ) grad d(o, ), u> motivates the introduction of a hyperbolic support function H of K. Hereby H() d(l(), D⁺()) is the distance of the K-supporting distance curve D⁺() from the line l() through the origin o with the direction angle . – The paper deals with the representation of C, s and k by H including extremal cases and an application of Minkowskis theorem to the characterization of circles by inequalities for their hyperbolic support function.