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If K is a closed convex cone, then K ∗∗ = K .
Next, show that C ∩ D ⊇ (C ∗ + D∗ )∗ and conclude (C ∩ D)∗ = C ∗ + D∗ . (d) Show that the dual of the polyhedral cone V = {x  Ax
V ∗ = {AT v  v 0} can be expressed as 0}. 1.6 Polar of a set. The polar of C ⊆ Rn is deﬁned as the set
C ◦ = {y ∈ Rn  y T x ≤ 1 for all x ∈ C }.
(a) Show that C ◦ is convex (even if C is not).
(b) What is the polar of a cone?
(c) What is the polar of the unit ball for a norm ·? (d) Show that if C is closed and convex, with 0 ∈ int C , then (C ◦ )◦ = C . 4 2 Convex functions 2.1 Maximum of a convex function over a polyhedron. Show that the maximum of a convex function f
over the polyhedron P = conv{v1 , . . . , vk } is achieved at one of its vertices, i.e.,
sup f (x) = max f (vi ). x∈P i=1,...,k (A stronger statement is: the maximum of a convex function over a closed bound convex set is
achieved at an extreme point, i.e., a point in the set that is not a convex combination of any other
points in the set.) Hint. Assume the statement is false, and use Jensen’s inequality.
2.2 A general vector composition rule...
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 Fall '13
 F.Borrelli
 The Aeneid

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