Lemma: Diagnostic comparison

The preceding networks share the operational properties of Theorem thm:network, but their standard generated cones have the properties in Table tab:diagnostic. Neither cone is strictly convex or boundary-transitive. The orthant is homogeneous but reducible. The diamond cone is irreducible but not homogeneous; its contrast with the Lorentz cone is illustrated in Figure fig:cones.

Proof. The operational assertions follow from the construction. Both cones have proper faces of dimension at least two. For the diamond cone, its four extreme generators can be written

Their relation space is one-dimensional, generated by $r_1+r_2-r_3-r_4=0$. A cone automorphism permutes the extreme rays. An automorphism fixing all four rays must scale them by the same positive factor, by uniqueness of that relation. Thus every orbit is contained in a finite union of rays after choosing representatives of the finite permutation action. It cannot be the three-dimensional interior.

A nontrivial decomposition of a three-dimensional proper cone has factor dimensions one and two. Such a product has three extreme rays, since a two-dimensional proper cone has two. The four-ray diamond cone is therefore irreducible.

Positive diagonal matrices act transitively on the interior of the orthant, which is a product of one-dimensional cones. For both examples, a boundary extreme-ray point cannot be mapped to a point in the relative interior of a higher-dimensional proper face. This proves failure of boundary transitivity.

For the standard Euclidean pairing, $K_{\diamond }^{\ast }=\{(t,y):t\ge \Vert y{\Vert }_{\infty }\}$, whereas the orthant is self-dual. The claim about the diamond pairing does not assert failure for every possible inner product.

Finally, a nondegenerate quadratic form in dimension at least three cannot vanish on an open subset of a hyperplane: polynomial restriction would make it vanish on the whole hyperplane, forcing a linear factor and rank at most two. The facets therefore preclude a nondegenerate quadratic boundary.