On Asymmetric Distances

Andrea C.G. Mennucci

Analysis and Geometry in Metric Spaces (2013)

  • Volume: 1, page 200-231
  • ISSN: 2299-3274

Abstract

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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.

How to cite

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Andrea C.G. Mennucci. "On Asymmetric Distances." Analysis and Geometry in Metric Spaces 1 (2013): 200-231. <http://eudml.org/doc/266882>.

@article{AndreaC2013,
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.},
author = {Andrea C.G. Mennucci},
journal = {Analysis and Geometry in Metric Spaces},
keywords = {Asymmetric metric; general metric; quasi metric; ostensible metric; Finsler metric; run–continuity; intrinsic metric; pathmetric; length structure; asymmetric metric; lenght structure; idempotency; induced semidistances},
language = {eng},
pages = {200-231},
title = {On Asymmetric Distances},
url = {http://eudml.org/doc/266882},
volume = {1},
year = {2013},
}

TY - JOUR
AU - Andrea C.G. Mennucci
TI - On Asymmetric Distances
JO - Analysis and Geometry in Metric Spaces
PY - 2013
VL - 1
SP - 200
EP - 231
AB - 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.
LA - eng
KW - Asymmetric metric; general metric; quasi metric; ostensible metric; Finsler metric; run–continuity; intrinsic metric; pathmetric; length structure; asymmetric metric; lenght structure; idempotency; induced semidistances
UR - http://eudml.org/doc/266882
ER -

References

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  13. [13] Gonçalo Gutierres and Dirk Hofmann. Approaching metric domains. Applied Categorical Structures, pages 1–34, (2012). Zbl1294.06009
  14. [14] H. Hopf and W. Rinow. Ueber den Begriff der vollständigen differentialgeometrischen Fläche. Comment. Math. Helv., 3(1):209–225, (1931). [Crossref] Zbl0002.35004
  15. [15] A. C. G. Mennucci. Regularity and variationality of solutions to Hamilton-Jacobi equations. part ii: variationality, existence, uniqueness. Applied Mathematics and Optimization, 63(2), (2011). [WoS] Zbl1216.49025
  16. [16] A. C. G. Mennucci. Geodesics in asymmetric metric spaces. In preparation, (2013). Zbl1310.53039
  17. [17] Athanase Papadopoulos. Metric spaces, convexity and nonpositive curvature, volume 6 of IRMA Lectures in Mathematics and Theoretical Physics. European Mathematical Society (EMS), Zürich, (2005). 
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