Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 133 / 4 / b

Let $p$ be the midpoint of a geodesic $[y_1,y_2]$. Since $Y$ is a convex subset of a geodesic metric space, $p\in Y$. In the geodesic triangle with vertices $x,y_1,y_2$, the point $p$ lies within $\delta$ of $[x,y_1]$ or $[x,y_2]$. By symmetry suppose $q\in [x,y_1]$ and $d(p,q)\le \delta$. Put $L=d(y_1,y_2)$. Thenand consequentlyBecause $y_1$ is a closest point of $Y$ to $x$ and $p\in Y$, $d(x,y_1)\le d(x,p)$. Combining the inequalities gives $L\le 4\delta$. This is the coarse uniqueness of a closest point in a hyperbolic metric space.