(Separation)
Let \(\sigma \subseteq N_\RR\) be a convex polyhedral cone and let \(v \in N_\RR\) with \(v \notin \sigma\). Then there is a support vector \(u \in \dualof{\sigma}\) with \[ \inp{u}{v} < 0 . \] That is, \(v\) lies strictly on the negative side of a hyperplane that has all of \(\sigma\) on its non-negative side.
(Double duality)
\[ \dualof{(\dualof{\sigma})} = \sigma . \]
The inclusion \(\sigma \subseteq \dualof{(\dualof{\sigma})}\) is the definition: every \(v \in \sigma\) pairs non-negatively with every \(u \in \dualof{\sigma}\).
For the reverse, take \(v \notin \sigma\) and apply separation to get \(u \in \dualof{\sigma}\) with \(\inp{u}{v} < 0\). That \(u\) witnesses \(v \notin \dualof{(\dualof{\sigma})}\).
(What separation rests on)
Separation is the finite-dimensional Hahn–Banach statement, and for a polyhedral cone it is elementary: \(\sigma\) is closed and convex, so the point of \(\sigma\) nearest to \(v\) exists, and the vector from it to \(v\) gives the required \(u\) after a sign change. Convexity and closedness are both needed, and polyhedral cones have both for free.
Double duality makes the cone-dual correspondence reversible. The equality \((\sigma^\vee)^\vee=\sigma\) means that \(\sigma\subseteq N_\RR\) and \(\sigma^\vee\subseteq M_\RR\) determine each other.
Dependencies
D-Q7Q2NFrom a fan to a toric varietyuses
Backlinks
PR-FULFACEDUALFaces of a cone and faces of its dual correspond contravariantlyusesPR-FULFACETFacet normals and the half-space presentation of a coneuses