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

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?

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

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

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 $\delta$-hyperbolic intrinsic metric space, for some $\delta>0$, then does $(X,d)$ admit a conical geodesic bicombing?

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