Minkowski versus Euclidean rank for products of metric spaces
نویسندگان
چکیده
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 does not hold in general.
منابع مشابه
The Asymptotic Rank of Metric Spaces
In this article we define and study a notion of asymptotic rank for metric spaces and show in our main theorem that for a large class of spaces, the asymptotic rank is characterized by the growth of the higher filling functions. For a proper, cocompact, simplyconnected geodesic metric space of non-curvature in the sense of Alexandrov the asymptotic rank equals its Euclidean rank.
متن کاملIsoperimetric Inequalities and the Asymptotic Rank of Metric Spaces
In this article we study connections between the asymptotic rank of a metric space and higher-dimensional isoperimetric inequalities. We work in the class of metric spaces admitting cone type inequalities which, in particular, includes all Hadamard spaces, i. e. simply connected metric spaces of nonpositive curvature in the sense of Alexandrov. As was shown by Gromov, spaces with cone type ineq...
متن کاملExperimental Evidence for Maximal Surfaces in a 3 Dimensional Minkowski Space
Conventional physical dogma, justified by the local success of Newtonian dynamics for particles, assigns a Euclidean metric with signature (plus, plus, plus) to the three spatial dimensions. Minimal surfaces are of zero mean curvature and negative Gauss curvature in a Euclidean space, which supports affine evolutionary processes. However, experimental evidence now indicates that the non-affine ...
متن کاملOn Areas and Integral Geometry in Minkowski Spaces
In a recent paper with J. A. Wieacker it was shown that certain integral-geometric formulas of Crofton type for lower-dimensional areas carry over from Euclidean to Minkowski spaces, if the Holmes-Thompson notion of Minkowskian area is employed and the Minkowski spaces are of a special type, called hyperme-tric. Here we show rst that there exist Minkowski spaces for which, among all axiomatical...
متن کاملDiameter-volume Inequalities and Isoperimetric Filling Problems in Metric Spaces
In this article we study metric spaces which admit polynomial diameter-volume inequalities for k-dimensional cycles. These generalize the notion of cone type inequalities introduced by M. Gromov in his seminal paper Filling Riemannian manifolds. In a first part we prove a polynomial isoperimetric inequality for k-cycles in such spaces, generalizing Gromov’s isoperimetric inequality of Euclidean...
متن کامل