Geodesic bicombing
In metric geometry, a geodesic bicombing distinguishes a class of geodesics of a metric space. The study of metric spaces with distinguished geodesics traces back to the work of the mathematician Herbert Busemann.[1] The convention to call a collection of paths of a metric space bicombing is due to William Thurston.[2] By imposing a weak global non-positive curvature condition on a geodesic bicombing several results from the theory of CAT(0) spaces and Banach space theory may be recovered in a more general setting.
Definition
Let be a metric space. A map is a geodesic bicombing if for all points the map is a unit speed metric geodesic from to , that is, , and for all real numbers .[3]
Different classes of geodesic bicombings
A geodesic bicombing is:
- reversible if for all and .
- consistent if whenever and .
- conical if for all and .
- convex if is a convex function on for all .
Examples
Examples of metric spaces with a conical geodesic bicombing include:
- Banach spaces.
- CAT(0) spaces.
- injective metric spaces.
- the spaces where is the first Wasserstein distance.
- any ultralimit or 1-Lipschitz retraction of the above.
Properties
- Every consistent conical geodesic bicombing is convex.
- Every convex geodesic bicombing is conical, but the reverse implication does not hold in general.
- Every proper metric space with a conical geodesic bicombing admits a convex geodesic bicombing.[3]
- Every complete metric space with a conical geodesic bicombing admits a reversible conical geodesic bicombing.[4]
References
- ^ Busemann, Herbert (1905-) (1987). Spaces with distinguished geodesics. Dekker. ISBN 0-8247-7545-7. OCLC 908865701.
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: CS1 maint: numeric names: authors list (link) - ^ Epstein, D. B. A. (1992). Word processing in groups. Jones and Bartlett Publishers. p. 84. ISBN 0-86720-244-0. OCLC 911329802.
- ^ a b Descombes, Dominic; Lang, Urs (2015). "Convex geodesic bicombings and hyperbolicity". Geometriae Dedicata. 177 (1): 367–384. doi:10.1007/s10711-014-9994-y. hdl:20.500.11850/87627. ISSN 0046-5755.
- ^ Basso, Giuliano; Miesch, Benjamin (2019). "Conical geodesic bicombings on subsets of normed vector spaces". Advances in Geometry. 19 (2): 151–164. arXiv:1604.04163. doi:10.1515/advgeom-2018-0008. hdl:20.500.11850/340286. ISSN 1615-7168. S2CID 15595365.