Geodesic bicombing explained
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
\sigma\colonX x X x [0,1]\toX
is a
geodesic bicombing if for all points
the map
\sigmaxy( ⋅ ):=\sigma(x,y, ⋅ )
is a unit speed metric geodesic from
to
, that is,
,
and
d(\sigmaxy(s),\sigmaxy(t))=\verts-t\vertd(x,y)
for all real numbers
.
[3] Different classes of geodesic bicombings
A geodesic bicombing
\sigma\colonX x X x [0,1]\toX
is:
- reversible if for all
and
.
- consistent if whenever
x,y\inX,0\leqs\leqt\leq1,p:=\sigmaxy(s),q:=\sigmaxy(t),
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:
where
is the first
Wasserstein distance.
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.
- Every complete metric space with a conical geodesic bicombing admits a reversible conical geodesic bicombing.[4]
Notes and References
- Book: Busemann, Herbert (1905-). Spaces with distinguished geodesics. 1987. Dekker. 0-8247-7545-7. 908865701.
- Book: Epstein, D. B. A.. Word processing in groups. 1992. Jones and Bartlett Publishers. 0-86720-244-0. 84. 911329802.
- Descombes. Dominic. Lang. Urs. 2015. Convex geodesic bicombings and hyperbolicity. Geometriae Dedicata. en. 177. 1. 367–384. 10.1007/s10711-014-9994-y. free. 0046-5755. 20.500.11850/87627. free.
- Basso. Giuliano. Miesch. Benjamin. 2019. Conical geodesic bicombings on subsets of normed vector spaces. Advances in Geometry. 19. 2. 151–164. 10.1515/advgeom-2018-0008. 1615-7168. 1604.04163. 20.500.11850/340286 . 15595365 .