face of a convex set, alternative definition of
Given any vectors and in , define the hyperplane by
note that this is the degenerate hyperplane if . As long as is nondegenerate, its removal disconnects . The upper halfspace of determined by is
A hyperplane is a supporting hyperplane for if its upper halfspace contains , that is, if .
Using this terminology, we can define a face of a convex set to be the intersection of with a supporting hyperplane of . Notice that we still get the empty set and as improper faces of .
Remarks. Let be a convex set.
If and are faces of intersecting in a point , then is a supporting hyperplane of , and . This shows that the faces of form a meet-semilattice.
Since each proper face lies on the base of the upper halfspace of some supporting hyperplane, each such face must lie on the relative boundary of .
|Title||face of a convex set, alternative definition of|
|Date of creation||2013-03-22 17:02:02|
|Last modified on||2013-03-22 17:02:02|
|Last modified by||mps (409)|