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.

Let ${\displaystyle (X,d)}$ be a metric space. A map ${\displaystyle \sigma \colon X\times X\times [0,1]\to X}$ is a geodesic bicombing if for all points ${\displaystyle x,y\in X}$ the map ${\displaystyle \sigma _{xy}(\cdot ):=\sigma (x,y,\cdot )}$ is a unit speed metric geodesic from ${\displaystyle x}$ to ${\displaystyle y}$, that is, ${\displaystyle \sigma _{xy}(0)=x}$, ${\displaystyle \sigma _{xy}(1)=y}$ and ${\displaystyle d(\sigma _{xy}(s),\sigma _{xy}(t))=\vert s-t\vert d(x,y)}$ for all real numbers ${\displaystyle s,t\in [0,1]}$.^[3]

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

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

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

Examples of metric spaces with a conical geodesic bicombing include:

• Banach spaces.
• CAT(0) spaces.
• injective metric spaces.
• the spaces ${\displaystyle (P_{1}(X),W_{1}),}$ where ${\displaystyle W_{1}}$ is the first Wasserstein distance.
• any ultralimit or 1-Lipschitz retraction of the above.

• 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]