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 [math]\displaystyle{ (X,d) }[/math] be a metric space. A map [math]\displaystyle{ \sigma\colon X\times X\times [0,1]\to X }[/math] is a geodesic bicombing if for all points [math]\displaystyle{ x,y\in X }[/math] the map [math]\displaystyle{ \sigma_{xy}(\cdot):=\sigma(x,y,\cdot) }[/math] is a unit speed metric geodesic from [math]\displaystyle{ x }[/math] to [math]\displaystyle{ y }[/math], that is, [math]\displaystyle{ \sigma_{xy}(0)=x }[/math], [math]\displaystyle{ \sigma_{xy}(1)=y }[/math] and [math]\displaystyle{ d(\sigma_{xy}(s), \sigma_{xy}(t))=\vert s-t\vert d(x,y) }[/math] for all real numbers [math]\displaystyle{ s,t\in [0,1] }[/math].[3]
Different classes of geodesic bicombings
A geodesic bicombing [math]\displaystyle{ \sigma\colon X\times X\times [0,1]\to X }[/math] is:
- reversible if [math]\displaystyle{ \sigma_{xy}(t)=\sigma_{yx}(1-t) }[/math] for all [math]\displaystyle{ x,y\in X }[/math] and [math]\displaystyle{ t\in [0,1] }[/math].
- consistent if [math]\displaystyle{ \sigma_{xy}((1-\lambda)s+\lambda t)=\sigma_{pq}(\lambda) }[/math] whenever [math]\displaystyle{ x,y\in X, 0\leq s\leq t\leq 1, p:=\sigma_{xy}(s), q:=\sigma_{xy}(t), }[/math]and [math]\displaystyle{ \lambda\in [0,1] }[/math].
- conical if [math]\displaystyle{ d(\sigma_{xy}(t), \sigma_{x^\prime y^\prime}(t))\leq (1-t)d(x,x^\prime)+t d(y,y^\prime) }[/math] for all [math]\displaystyle{ x,x^\prime, y, y^\prime\in X }[/math] and [math]\displaystyle{ t\in [0,1] }[/math].
- convex if [math]\displaystyle{ t\mapsto d(\sigma_{xy}(t), \sigma_{x^\prime y^\prime}(t)) }[/math] is a convex function on [math]\displaystyle{ [0,1] }[/math] for all [math]\displaystyle{ x,x^\prime, y, y^\prime\in X }[/math].
Examples
Examples of metric spaces with a conical geodesic bicombing include:
- Banach spaces.
- CAT(0) spaces.
- injective metric spaces.
- the spaces [math]\displaystyle{ (P_1(X),W_1), }[/math] where [math]\displaystyle{ W_1 }[/math] 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. http://worldcat.org/oclc/908865701.
- ↑ Epstein, D. B. A. (1992). Word processing in groups. Jones and Bartlett Publishers. pp. 84. ISBN 0-86720-244-0. OCLC 911329802. http://worldcat.org/oclc/911329802.
- ↑ 3.0 3.1 Descombes, Dominic; Lang, Urs (2015). "Convex geodesic bicombings and hyperbolicity" (in en). Geometriae Dedicata 177 (1): 367–384. doi:10.1007/s10711-014-9994-y. ISSN 0046-5755.
- ↑ Basso, Giuliano; Miesch, Benjamin (2019). "Conical geodesic bicombings on subsets of normed vector spaces". Advances in Geometry 19 (2): 151–164. doi:10.1515/advgeom-2018-0008. ISSN 1615-7168. https://www.degruyter.com/view/j/advgeom.2019.19.issue-2/advgeom-2018-0008/advgeom-2018-0008.xml.
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