Corollary: Exclusion of the polyhedral realizations

The standard generated representations of Propositions prop:ball and prop:orthant, and every invertible affine change of either representation, fail $\mathsf{S}$. More generally, an exact generated representation with a finite edge-increment set cannot satisfy $\mathsf{S}$ in dimension at least three.

Proof. Both standard cones fail strict convexity. The orthant additionally fails irreducibility, while $K_{\diamond }$ fails interior homogeneity (Lemma lem:diagnostic). These properties are invariant under linear isomorphism, and an affine change transforms displacement vectors by its linear part. The final statement follows from Proposition prop:finitestencil and Theorem thm:representation.