# On Asymmetric Distances

Analysis and Geometry in Metric Spaces (2013)

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

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topAndrea 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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