Published Paper
Inserted: 29 apr 2004
Last Updated: 7 jan 2015
Journal: Analysis and Geometry in Metric Spaces
Volume: 1
Pages: 200-231
Year: 2013
Doi: 10.2478/agms-2013-0004
Notes:
The older 2004 and 2007 versions contained an important mistake. Sorry. A new 2012 version is provided. In correcting the mistake a whole new part of the theory was discovered, so the material was divided in two papers (the second part is in http://cvgmt.sns.it/paper/2264/ ).
"Analysis and Geometry in Metric Spaces" is an open access paper: if interested, please download the paper from that source. The DOI link will bring you to the correct web page.
Abstract:
In this paper we discuss asymmetric length structure and asymmetric metric spaces.
A length structure induces a (semi)distance function; by using the total variation formula, a (semi)distance function induces a length. In the first part we identify a topology in the set of paths that best describes when the above operations are idempotent. As a typical application, we consider the length of paths defined by a Finslerian functional in Calculus of Variations.
In the second part we generalize the setting of \emph{General metric spaces} of Busemann, and discuss the newly found aspects of the theory: we identify three interesting classes of paths, and compare them; we note that a ``geodesic segment'' (as defined by Busemann) is not necessarily continuous in our setting; hence we present three different notions of ``intrinsic metric space''.
Keywords: Finsler metrics, metric space, 53C22, 53C60, quasi metric, intrinsic metric
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