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.
@article{bwmeta1.element.doi-10_2478_agms-2013-0004,
author = {Andrea C.G. Mennucci},
title = {On Asymmetric Distances},
journal = {Analysis and Geometry in Metric Spaces},
volume = {1},
year = {2013},
pages = {200-231},
zbl = {1293.53081},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.doi-10_2478_agms-2013-0004}
}
Andrea C.G. Mennucci. On Asymmetric Distances. Analysis and Geometry in Metric Spaces, Tome 1 (2013) pp. 200-231. http://gdmltest.u-ga.fr/item/bwmeta1.element.doi-10_2478_agms-2013-0004/
[1] D. Bao, S. S. Chern, and Z. Shen. An Introduction to Riemann-Finsler Geometry. (Springer–Verlag), (2000). | Zbl 0954.53001
[2] Leonard M. Blumenthal. Theory and applications of distance geometry. Second edition. Chelsea Publishing Co., New York, (1970). | Zbl 0050.38502
[3] V. I. Bogachev. Measure theory. Vol. I, II. Springer-Verlag, Berlin, (2007). | Zbl 1120.28001
[4] Dmitri Burago, Yuri Burago, and Sergei Ivanov. A course in metric geometry, volume 33 of Graduate Studies in Mathematics. American Mathematical Society, Providence, RI, (2001). | Zbl 1232.53037
[5] H. Busemann. Local metric geometry. Trans. Amer. Math. Soc., 56:200–274, (1944). | Zbl 0061.37403
[6] H. Busemann. The geometry of geodesics, volume 6 of Pure and applied mathematics. Academic Press (New York), (1955).
[7] H. Busemann. Recent synthetic differential geometry, volume 54 of Ergebnisse der Mathematik und ihrer Grenzgebiete. Springer Verlag, (1970). | Zbl 0194.53701
[8] G. Buttazzo. Semicontinuity, relaxation and integral representation in the calculus of variation. Number 207 in scientific & technical. Longman, (1989).
[9] S. Cohn-Vossen. Existenz kürzester Wege. Compositio math., Groningen,, 3:441–452, (1936). | Zbl 62.0861.01
[10] Bernard Dacorogna. Direct methods in the calculus of variations, volume 78 of Applied Mathematical Sciences. Springer, New York, second edition, (2008). | Zbl 1140.49001
[11] Alessandro Duci and Andrea Mennucci. Banach-like metrics and metrics of compact sets. eprint arXiv:0707.1174. (2007).
[12] M. Gromov. Metric Structures for Riemannian and Non-Riemannian Spaces, volume 152 of Progress in Mathematics. Birkhäuser Boston, (2007). | Zbl 1113.53001
[13] Gonçalo Gutierres and Dirk Hofmann. Approaching metric domains. Applied Categorical Structures, pages 1–34, (2012). | Zbl 1294.06009
[14] H. Hopf and W. Rinow. Ueber den Begriff der vollständigen differentialgeometrischen Fläche. Comment. Math. Helv., 3(1):209–225, (1931). [Crossref] | Zbl 0002.35004
[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] | Zbl 1216.49025
[16] A. C. G. Mennucci. Geodesics in asymmetric metric spaces. In preparation, (2013). | Zbl 1310.53039
[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).
[18] B. B. Phadke. Nonsymmetric weakly complete g-spaces. Fundamenta Mathematicae, (1974). | Zbl 0269.53029
[19] E. M. Zaustinsky. Spaces with non-symmetric distances. Number 34 in Mem. Amer. Math. Soc. AMS, (1959). | Zbl 0115.39802