Theorem: Conditional Lorentz classification

Let $K\subset V$ be a homogeneous, strictly convex proper cone, with $\dim V=m\ge 3$. Then there is an invertible linear map $T:V\to {\mathbb{R}}^m$ such that

This is the classification consequence recorded as En08-A thm:weaker. Homogeneity and strict convexity are additional hypotheses, not consequences of $\mathsf{S}$.

Proof. Apply Ito and Lourenço's classification of strictly convex homogeneous cones ItoLourenco (Theorem 7). Their lower-dimensional alternatives are excluded by $\dim V\ge 3$. The remaining alternative is a Lorentz cone. The classification is not re-proved here; its operational use is the conditional statement recorded in En08-A CheungEn08A (thm:weaker).