A velocity-set physical model

Let $K_{\mathrm{phys}}(x,t)\subset {\mathbb{R}}^d$ be a declared compact convex velocity set. Assume that the paths used in a certificate are absolutely continuous and satisfy

Define $p{⪯}_{K}q$ when an admissible path joins the events with nondecreasing time. Concatenation makes this relation transitive.

For constant $K$, the reachable displacement set in time $T>0$ is exactly $TK$: averages of allowed velocities lie in $K$, and a constant velocity realizes every point of $TK$.