Steiner Ratio for Hadamard Surfaces of Curvature at Most k < 0

Abstract

In this paper, the Hadamard surfaces of curvature at most k are investigated, which are a particular case of Alexandrov surfaces. It was shown that the total angle at the points of an Hadamard surface is not less than 2π. The Steiner ratio of an Hadamard surface was obtained for the case where the surface is unbounded and k < 0.

Branched coverings and Steiner ratio

“…At present, an exact value of the Steiner ratio is calculated for the Manhattan plane (Hwang, ), Lobachevski plane (Innami and Kim, ), and Hadamard spaces (i.e., simply connected A. D. Alexandrov spaces) of negative curvature (Zaval'nyuk, ). Also, there are several estimates and theorems describing some properties of this interesting characteristic of metric spaces, see a review in Cieslik (, ).…”