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We give new characterizations of the asymptotic cone and the lineality space of a convex set; to this effect we use the notions of margin, extreme point, face and cone of admissible directions. For example we find this statement: if A. is a closed, line-free and finite-dimensional convex set, then wherepis the set of extreme points of A and W(A,α) is the witness cone of A from the point α.
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