A M\"obius Characterization of Metric Spheres

In this paper we characterize compact extended Ptolemy metric spaces with many circles up to M\"obius equivalence. This characterization yields a M\"obius characterization of the $n$-dimensional spheres $S^n$ and hemispheres $S^n_+$ when endowed with their chordal metrics. In particular, we show that every compact extended Ptolemy metric space with the property that every three points are contained in a circle is M\"obius equivalent to $(S^n,d_0)$ for some $n\ge 1$, the $n$-dimensional sphere $S^n$ with its chordal metric.Comment: 24 pages, 1 figur

Products of hyperbolic metric spaces

Let (X_i,d_i), i=1,2, be proper geodesic hyperbolic metric spaces. We give a general construction for a ``hyperbolic product'' X_1{times}_h X_2 which is itself a proper geodesic hyperbolic metric space and examine its boundary at infinity.Comment: 17 page

Minkowski- versus Euclidean rank for products of metric spaces

We introduce a notion of the Euclidean- and the Minkowski rank for arbitrary metric spaces and we study their behaviour with respect to products. We show that the Minkowski rank is additive with respect to metric products, while additivity of the Euclidean rank only holds under additional assumptions, e.g. for Riemannian manifolds. We also study products with nonstandard product metrics.Comment: 20 pages, 1 figur

Non Standard Metric Products

We consider a fairly general class of natural non standard metric products and classify those amongst them, which yield a product of certain type (for instance an inner metric space) for all possible choices of factors of this type (inner metric spaces). We further prove the additivity of the Minkowski rank for a large class of metric products.Comment: 13 pages, This paper extends the results of the second part of arXiv math.MG/0102107. Note that the first part of arXiv math.MG/0102107 has been published in Adv. Geom. 2 (2002), 123-13