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In metric geometry, a geodesic bicombing distinguishes a class of geodesics of a metric space.

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, $\sigma (0)=x$, $\sigma _{xy}(1)$ 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 $\ldots$

• 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 the function $t\mapsto d(\sigma _{xy}(t),\sigma _{x^{\prime }y^{\prime }}(t))$ is a convex function on $[0,1]$ for all $x,x^{\prime },y,y^{\prime }\in X$.
• consistent if $\sigma _{pq}(\lambda )=\sigma _{xy}((1-\lambda )s+\lambda t)$ whenever $x,y\in X,0\leq s\leq t\leq 1,p:=\sigma _{xy}(s),q:=\sigma _{xy}(t),$ and $\lambda \in [0,1]$.
• reversible if $\sigma _{xy}(t)=\sigma _{yx}(1-t)$ for all $x,y\in X$ and $t\in [0,1]$.

Examples

Examples of metric spaces with a conical geodesic bicombing include

• Banach spaces
• CAT(0) spaces
• injective metric spaces
• any ultralimit or 1-Lipschitz retraction of the above

Properties

• Every consistent conical geodesic bicombing is convex.
• 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.