Metrics on hyperbolic convex polyhedra
There is a traditional way of defining a metric for the projective model of hyperbolic space: For a unit ball \(C \subset R^n\) given two interior points \(a,b\) one considers the usual line through \(a,b\) and finds the intersections \(x,y\) of this line with the boundary of \(C\). The Hilbert distance between \(a\) and \(b\) is simply \(-\frac{1}{2} \log(\frac{xa}{xb} \frac{by}{ay})\). Thinking of \(\partial C\) as a quadric not as a convex set one can generalize the above formula for points in \(R^n -C\). The ``distance'' becomes then a certain complex number. One can for example speak of the distance between two points outside \(C\). The new space with this ``distance'' is called the Sitter hyperbolic space, denoted \(HS^n\) and it has natural notions of geodesic and polyhedron. Sitter spaces have also a notion of duality between points and hyperplanes. The main goal of this article is to show that several well-known results about polyhedra in the classical hyperbolic space extend to Sitter spaces. The author presents generalizations of results by Alexandrov, Andreev, Pogorelov and Rivin.
scientific article; zbMATH DE number 1203424