Barbier’s theorem in the Lobachevski plane

Abstract

In the Lobachevski plane, horocycles with the same center are geodesic parallels and are natural replacements for the lines used in defining the support function of a convex curve and the notion of constant width in the Euclidean plane. In this paper, analogs based on horocycles are obtained for Christoffel’s formula, which expresses the radius of curvature of a convex curve in terms of its support function, and Barbier’s theorem, which relates the length and width of a convex curve of constant width.