# Do Gromov hyperbolic spaces admit concical geodesic bicombings?

Consider a metric space $$(X,d)$$ with a distinguished selection of geodesics, i.e. a geodesic bicombing $$\sigma:X\times X\times [0,1]\rightarrow X$$. We call a geodesic bicombing conical if it satisfies the following "convexity-type condition" $$d(\sigma_{xy}(t), \sigma_{x^\prime y^\prime}(t))\leq (1-t)d(x,x^\prime)+t d(y,y^\prime)$$ for all $$x,x',y,y'\in X$$ and $$t\in [0,1]$$.

If $$(X,d)$$ is a complete, separable, contractible Gromov ($$\delta$$) hyperbolic intrinsic metric space, for some $$\delta>0$$, then does $$(X,d)$$ admit a conical geodesic bicombing?

• This condition implies contractibility of $X$, are you sure that you really mean it? Sep 18 at 16:28
• I think you should write "Gromov-hyperbolic" rather than $\delta$-hyperbolic. $\delta$ is a parameter, which does not play any role in your question.
– YCor
Sep 18 at 16:29
• @MoisheKohan Ah, in the case I had in mind, my space is a contracible ANR which is additionally $\delta$-hyperbolic. Otherwise, yes, I would not have a shot. Sep 18 at 16:47
• Ok, there is still an easy counter-example: take the unit sphere with the standard (Riemannian) metric and remove a point from it. Maybe you are making further assumptions to rule out such examples. Note that metric completeness is not enough (remove a small open disk from the unit sphere). Sep 18 at 16:54
• As I explained, the added assumptions are insufficient. Maybe you should explain in detail the actual example you are interested in. Sep 18 at 17:33

None of your extra assumptions will suffice. An easy example is obtained as follows. Start with the unit 2-sphere $$S^2$$ with the standard Riemannian metric $$ds^2$$ induced from the embedding in $${\mathbb R}^3$$. Remove from $$S^2$$ a small open metric disk (of radius $$<\pi/2$$, cetered at a point $$p\in S^2$$). Equip the resulting space $$X$$ with the length structure coming from $$ds^2$$. The corresponding metric $$d$$ on $$X$$ will be complete, etc, since $$X$$ is compact. However, it is clear that $$(X,d)$$ fails the required convexity property: Take the point $$x=-p$$. Then every $$y\in (X,d)$$ is connected to $$x$$ by a unique geodesic. But the convexity property fails for such geodesics (just look at $$y, y'$$ which belong to the boundary circle of $$X$$).