Theorem: Fixed-periodic-model isotropy obstruction

Let $D\ge 2$. A fixed finite periodic transport model, rescaled with a fixed length-to-time ratio, cannot have a nonzero Euclidean ball as its limiting scheduled velocity set.

More generally, a family of compact velocity polytopes with uniformly bounded numbers of vertices cannot converge in Hausdorff distance to a nonzero Euclidean ball.

Proof. For the fixed model, uniform length and time rescaling leaves (eq:velocitypolytope) unchanged. It is a finite polytope. A nonzero Euclidean ball in $D\ge 2$ has infinitely many extreme points and is not a finite polytope.

For the second statement, let the vertex count be bounded by $M$. Pad each vertex list to length $M$ by repetition. Hausdorff convergence to a compact ball bounds all vertices. A subsequence has coordinatewise convergent vertex lists. Its limiting convex hull is a polytope with at most $M$ vertices and also equals the asserted Hausdorff limit, a contradiction.