Geodesics in Asymmetic Metric Spaces

Andrea C. G. Mennucci

Analysis and Geometry in Metric Spaces (2014)

  • Volume: 2, Issue: 1, page 115-153, electronic only
  • ISSN: 2299-3274

Abstract

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In a recent paper [17] we studied asymmetric metric spaces; in this context we studied the length of paths, introduced the class of run-continuous paths; and noted that there are different definitions of “length spaces” (also known as “path-metric spaces” or “intrinsic spaces”). In this paper we continue the analysis of asymmetric metric spaces.We propose possible definitions of completeness and (local) compactness.We define the geodesics using as admissible paths the class of run-continuous paths.We define midpoints, convexity, and quasi-midpoints, but without assuming the space be intrinsic.We distinguish all along those results that need a stronger separation hypothesis. Eventually we discuss how the newly developed theory impacts the most important results, such as the existence of geodesics, and the renowned Hopf-Rinow (or Cohn-Vossen) theorem.

How to cite

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Andrea C. G. Mennucci. "Geodesics in Asymmetic Metric Spaces." Analysis and Geometry in Metric Spaces 2.1 (2014): 115-153, electronic only. <http://eudml.org/doc/267188>.

@article{AndreaC2014,
abstract = {In a recent paper [17] we studied asymmetric metric spaces; in this context we studied the length of paths, introduced the class of run-continuous paths; and noted that there are different definitions of “length spaces” (also known as “path-metric spaces” or “intrinsic spaces”). In this paper we continue the analysis of asymmetric metric spaces.We propose possible definitions of completeness and (local) compactness.We define the geodesics using as admissible paths the class of run-continuous paths.We define midpoints, convexity, and quasi-midpoints, but without assuming the space be intrinsic.We distinguish all along those results that need a stronger separation hypothesis. Eventually we discuss how the newly developed theory impacts the most important results, such as the existence of geodesics, and the renowned Hopf-Rinow (or Cohn-Vossen) theorem.},
author = {Andrea C. G. Mennucci},
journal = {Analysis and Geometry in Metric Spaces},
keywords = {asymmetric metric; general metric; quasi metric; ostensible metric; Finsler metric; path metric; length space; geodesic curve; Hopf-Rinow theorem; intrinsic space},
language = {eng},
number = {1},
pages = {115-153, electronic only},
title = {Geodesics in Asymmetic Metric Spaces},
url = {http://eudml.org/doc/267188},
volume = {2},
year = {2014},
}

TY - JOUR
AU - Andrea C. G. Mennucci
TI - Geodesics in Asymmetic Metric Spaces
JO - Analysis and Geometry in Metric Spaces
PY - 2014
VL - 2
IS - 1
SP - 115
EP - 153, electronic only
AB - In a recent paper [17] we studied asymmetric metric spaces; in this context we studied the length of paths, introduced the class of run-continuous paths; and noted that there are different definitions of “length spaces” (also known as “path-metric spaces” or “intrinsic spaces”). In this paper we continue the analysis of asymmetric metric spaces.We propose possible definitions of completeness and (local) compactness.We define the geodesics using as admissible paths the class of run-continuous paths.We define midpoints, convexity, and quasi-midpoints, but without assuming the space be intrinsic.We distinguish all along those results that need a stronger separation hypothesis. Eventually we discuss how the newly developed theory impacts the most important results, such as the existence of geodesics, and the renowned Hopf-Rinow (or Cohn-Vossen) theorem.
LA - eng
KW - asymmetric metric; general metric; quasi metric; ostensible metric; Finsler metric; path metric; length space; geodesic curve; Hopf-Rinow theorem; intrinsic space
UR - http://eudml.org/doc/267188
ER -

References

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