= Solution
A subset is a <quasiconvex subset> with constant $C\geq0$ if every <metric geodesic> segment with endpoints in it lies within distance $C$ of it. Explicitly,
$$
\boxed{p,q\in N,\ z\in[p,q]\quad\Longrightarrow\quad d_M(z,N)\leq C,}
$$
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 $C\geq0$. 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.
Back to article page
