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Unformatted text preview: at happens on the boundaries of these cones,
so you really needn’t worry about it. But we make some comments here for those who do care
about such things.
The cone Kexp as deﬁned above is not closed. To obtain its closure, we need to add the points
{(x, y, z )  x ≤ 0, y = 0, z ≥ 0}.
(This makes no diﬀerence, since the dual of a cone is equal to the dual of its closure.)
1.5 Dual of intersection of cones. Let C and D be closed convex cones in Rn . In this problem we will
show that
(C ∩ D )∗ = C ∗ + D ∗ .
Here, + denotes set addition: C ∗ + D∗ is the set {u + v  u ∈ C ∗ , v ∈ D∗ }. In other words, the
dual of the intersection of two closed convex cones is the sum of the dual cones.
3 (a) Show that C ∩ D and C ∗ + D∗ are convex cones. (In fact, C ∩ D and C ∗ + D∗ are closed, but
we won’t ask you to show this.)
(b) Show that (C ∩ D)∗ ⊇ C ∗ + D∗ . (c) Now let’s show (C ∩ D)∗ ⊆ C ∗ + D∗ . You can do this by ﬁrst showing
(C ∩ D)∗ ⊆ C ∗ + D∗ ⇐⇒ C ∩ D ⊇ (C ∗ + D∗ )∗ .
You can use the following...
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This note was uploaded on 09/10/2013 for the course C 231 taught by Professor F.borrelli during the Fall '13 term at University of California, Berkeley.
 Fall '13
 F.Borrelli
 The Aeneid

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