Salient Convex Cones and Perfect Half-spaces
A convex cone is said to be flat if it contains some nonzero vector x and its opposite -x; and salient otherwise.
A blunt convex cone is necessarily salient, but the converse is not necessarily true. A convex cone C is salient if and only if C(-C){0}; that is, if and only if C does not contain any non-trivial linear subspace of V.
A perfect half-space of V is defined recursively as follows: if V is zero-dimensional, then it is the set {0}, else it is any open half-space H of V, together with a perfect half-space of the bounding hyperplane of H.
Every perfect half-space is a salient convex cone; and, moreover, every salient convex cone is contained in a perfect half-space. In other words, the perfect half-spaces are the maximal salient convex cones (under the containment order). In fact, it can be proved that every pointed salient convex cone (independently of whether it is topologically open, closed, or mixed) is the intersection of all the perfect half-spaces that contain it.
Read more about this topic: Convex Cone
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