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A subset is a quasiconvex subset with constant $C\ge 0$ if every metric geodesic segment with endpoints in it lies within distance $C$ of it. Explicitly,for every choice of the metric geodesic $[p,q]$. The distance is computed in the ambient geodesic metric space, not in the intrinsic path metric of $N$. One only asks the metric geodesic to stay close to $N$, rather than requiring it to lie in $N$; the latter makes $N$ a convex subset of a geodesic metric space.

The usual convention takes $C\ge 0$. If the printed allowance $C\in \mathbb{R}$ is interpreted literally using the distance inequality, a negative $C$ cannot work for a nonempty $N$, since the constant metric geodesic at a point of $N$ has distance zero. For the empty subset the endpoint condition is vacuous.

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.