Manuscript En08-A@1 · En08-A.tex · definition · c688d76d

Definition: Faces and strict convexity of a cone

Signature and Lorentz-cone selection › The symmetric-cone representation theorem · lines 1297–1311

En08-A@1#def:strict, En08-A@1#eq:strict

A face of $K$ is a convex subcone $F \subseteq K$ such that $u, v \in K$ and $u + v \in F$ imply $u, v \in F$ . It is proper when $F \neq = K$ . A proper cone is strictly convex if

u, v \in K ∖ {0}, u \in / R_{> 0} v ⟹ u + v \in int K .

Equivalently, every nonzero proper face is an extreme ray.

Displayed formulas in this passage

En08-A@1#eq:strict · line 1303
$u, v \in K ∖ {0}, u \in / R_{> 0} v ⟹ u + v \in int K .$

Nearby in this section

section: The symmetric-cone representation theorem
assumption: Assumption: Symmetric-cone selection axiom
section: The symmetric-cone representation theorem
theorem: Theorem: Representation from cone order and
corollary: Corollary: Application to records
corollary: Corollary: Exclusion of the polyhedral realizations
section: The symmetric-cone representation theorem
figure: Figure