In this chapter, we prove the Separation Theorem for convex sets and extract from it basic results on the geometry of closed convex sets, specifically (a) supporting hyperplanes, extreme points, and the (finite-dimensional) Krein--Milman Theorem, (b) recessive directions and recessive cone of a convex set, (c) the definition and basic properties of the dual cone, (d) the (finite-dimensional) Dubovitski--Milutin Lemma, (e) existence of bases and extreme rays of nontrivial closed pointed cones and their relation to extreme points of the cone’s base, (f)the (finite-dimensional) Krein--Milman Theorem in conic form, and (g) polarity.
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