Proposition: Velocity polytope and finite-time offset

With $q=|V(Q)|$ and $D_{\max }={\max }_e\Vert {\Delta }_e{\Vert }_2$, every path completed within time $u$ satisfies

Its limiting velocity set is $K_{\mathsf{Sch}}$.

Proof. Remove cycles from a walk until the remainder has no repeated vertex. It has at most $q-1$ edges. The removed cycles decompose into simple cycles, and their total displacement is a duration-weighted sum of their velocities. Since $0\in K_{\mathsf{Sch}}$, that contribution belongs to $u{K}_{\mathsf{Sch}}$.

Conversely, repeat cycles in desired time proportions and use fixed connector walks, available by strong connectivity. Connector costs are negligible along increasingly long repetitions; waiting realizes the zero contribution. This approximates every convex combination.