Convex subset of a geodesic metric space

A subset $Y$ of a geodesic metric space is convex when every geodesic whose endpoints lie in $Y$ is contained in $Y$.

If $Y$ is convex in a $\delta$-hyperbolic metric space and two points of $Y$ both minimize distance from $x$, then their distance is at most $4\delta$. Thus a closest-point projection to $Y$, when it exists, is unique up to uniformly bounded error.