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Analysis and Geometry in Metric Spaces

Ed. by Ritoré, Manuel


CiteScore 2017: 0.65

SCImago Journal Rank (SJR) 2017: 1.063
Source Normalized Impact per Paper (SNIP) 2017: 0.833

Mathematical Citation Quotient (MCQ) 2017: 0.86

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Online
ISSN
2299-3274
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On Asymmetric Distances

Andrea C.G. Mennucci
Published Online: 2013-06-11 | DOI: https://doi.org/10.2478/agms-2013-0004

Abstract

In this paper we discuss asymmetric length structures 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 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: Asymmetric metric; general metric; quasi metric; ostensible metric; Finsler metric; run–continuity; intrinsic metric; pathmetric; length structure

MSC: 54C99; 54E25; 26A45

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About the article

Received: 2013-03-24

Accepted: 2013-05-15

Published Online: 2013-06-11


Citation Information: Analysis and Geometry in Metric Spaces, Volume 1, Pages 200–231, ISSN (Online) 2299-3274, DOI: https://doi.org/10.2478/agms-2013-0004.

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