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

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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

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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

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Examples of metric spaces with a conical geodesic bicombing include:

Properties

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  • 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

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  1. ^ Busemann, Herbert (1905-) (1987). Spaces with distinguished geodesics. Dekker. ISBN 0-8247-7545-7. OCLC 908865701.{{cite book}}: CS1 maint: numeric names: authors list (link)
  2. ^ Epstein, D. B. A. (1992). Word processing in groups. Jones and Bartlett Publishers. p. 84. ISBN 0-86720-244-0. OCLC 911329802.
  3. ^ 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.
  4. ^ 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.