Proposition: Boundary transitivity as a sufficient replacement

For a proper cone $K$ of dimension at least two, if $\mathrm{Aut}(K)$ acts transitively on $\partial K\setminus \{0\}$, then $K$ is strictly convex. Consequently, in Theorem thm:representation its strict-convexity clause may be replaced by boundary transitivity.

Proof. Choose a compact base of $K$ using a strictly positive linear functional. An extreme point of that base gives an extreme ray, which lies on the boundary in dimension at least two. Linear cone automorphisms preserve extremality. Transitivity therefore makes every nonzero boundary point generate an extreme ray.

If nonproportional $u,v\in K\setminus \{0\}$ had $u+v\in \partial K$, that boundary ray would not be extreme. This contradicts the preceding conclusion.