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. The convention to call a collection of paths of a metric space bicombing is due to William Thurston. 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 $(X,d)$ be a metric space. A map $\sigma :X\times X\times [0,1]\to X$ is a geodesic bicombing if for all points $x,y\in X$ the map ${\sigma }_{xy}(\cdot ):=\sigma (x,y,\cdot )$ is a unit speed metric geodesic from $x$ to $y$, that is, ${\sigma }_{xy}(0)=x$, ${\sigma }_{xy}(1)=y$ and $d({\sigma }_{xy}(s),{\sigma }_{xy}(t))=|s-t|d(x,y)$ for all real numbers $s,t\in [0,1]$.

Different classes of geodesic bicombings

A geodesic bicombing $\sigma :X\times X\times [0,1]\to X$ is:

• reversible if $${\sigma }_{xy}(t)={\sigma }_{yx}(1-t)$$ for all $x,y\in X$ and $t\in [0,1]$.
• consistent if $${\sigma }_{xy}((1-\lambda )s+\lambda t)={\sigma }_{pq}(\lambda )$$ whenever $x,y\in X,0\le s\le t\le 1,p:={\sigma }_{xy}(s),q:={\sigma }_{xy}(t),$and $\lambda \in [0,1]$.

• conical if $$d({\sigma }_{xy}(t),{\sigma }_{x^{\prime }{y}^{\prime }}(t))\le (1-t)d(x,x^{\prime })+td(y,y^{\prime })$$ for all $x,x^{\prime },y,y^{\prime }\in X$ and $t\in [0,1]$.
• convex if $$t\mapsto d({\sigma }_{xy}(t),{\sigma }_{x^{\prime }{y}^{\prime }}(t))$$ is a convex function on $[0,1]$ for all $x,x^{\prime },y,y^{\prime }\in X$.

Examples

Examples of metric spaces with a conical geodesic bicombing include:

• Banach spaces.
• CAT(0) spaces.
• injective metric spaces.
• the spaces $(P_1(X),W_1),$ where $W_1$ 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.
• Every complete metric space with a conical geodesic bicombing admits a reversible conical geodesic bicombing.