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 $$\sigma\colon X\times X\times [0,1]\to X$$ 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 $$d(\sigma_{xy}(s), \sigma_{xy}(t))=\vert s-t\vert d(x,y)$$ for all real numbers $$s,t\in [0,1]$$.

Different classes of geodesic bicombings
A geodesic bicombing $$\sigma\colon X\times X\times [0,1]\to X$$ 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 $$x,y\in X, 0\leq s\leq t\leq 1, p:=\sigma_{xy}(s), q:=\sigma_{xy}(t), $$and $$\lambda\in [0,1]$$.


 * conical if $$d(\sigma_{xy}(t), \sigma_{x^\prime y^\prime}(t))\leq (1-t)d(x,x^\prime)+t d(y,y^\prime)$$ 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.