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
  $$d(\sigma _{xy}(t),\sigma _{x^{\prime }y^{\prime }}(t))\leq (1-t)d(x,x^{\prime })+td(y,y^{\prime })$$
  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.