Coarse uniqueness of a closest point in a hyperbolic metric space

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.

If $\gamma :\mathbb{R}\to X$ is an isometrically embedded geodesic line, then every $x\in X$ has a closest point on $\gamma$. The function $t\mapsto d(x,\gamma (t))$ is continuous and tends to infinity with $|t|$. Any two minimizing parameters differ by at most $4\delta$, and therefore by at most $6\delta$, in a $\delta$-hyperbolic metric space.