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