Manuscript XXII-C@1 · XXII-C.tex · section · b9a05a24

Classification properties: definitions, not applications

The six extraction obligations › Classification properties: definitions, not applications · lines 1093–1125

XXII-C@1#eq:homogeneity, XXII-C@1#eq:strict

For a constructed proper cone $K_{D} \subset R^{m}$ , homogeneity means

Aut (K_{D}) = {A \in GL (R^{m}) : A (K_{D}) = K_{D}}, \forall u, v \in int K_{D} \exists A \in Aut (K_{D}) : A u = v .

Cone strict convexity means

u, v \in K_{D} ∖ {0}, u, v non-proportional ⟹ u + v \in int K_{D} .

Ito and Lourenço classify strictly convex homogeneous cones ItoLourenco (Theorem 7). XXII-A records the resulting conditional Lorentz classification in dimension at least three CheungXXIIA (thm:classification). This note states no application of that result to a cone obtained from $E_{m}$ , because no such selected cone has been constructed.

Complete positivity is an operator-cone property. Its relation to the event-difference cone (eq:generated) is not an identity. Likewise, a scheduler interleaving is not automatically an element of $Aut (K_{D})$ .

Displayed formulas in this passage

XXII-C@1#eq:homogeneity · line 1096
$Aut (K_{D}) = {A \in GL (R^{m}) : A (K_{D}) = K_{D}}, \forall u, v \in int K_{D} \exists A \in Aut (K_{D}) : A u = v .$
XXII-C@1#eq:strict · line 1104
$u, v \in K_{D} ∖ {0}, u, v non-proportional ⟹ u + v \in int K_{D} .$