Calculus of Variations and Geometric Measure Theory

A. C. G. Mennucci

On asymmetric distances

created on 29 Apr 2004
modified by mennucci on 07 Jan 2015


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

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 ).

"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.


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