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Variational methods - properties of convex hull

  1. Mar 3, 2008 #1
    Show the following properties of convex hull:
    (a) Co(CoA) = Co(A)
    (b) Co(AUB) [tex]\supseteq[/tex]Co(A) U Co(B)
    (c) If A[tex]\subseteq[/tex]B then Co(AUB)=Co(B)
    (d) If A[tex]\subseteq[/tex]B then Co(A)[tex]\subseteq[/tex]Co(B)

    The definition of a convex hull is a set of points A is the minimum convex set containing A.
    (c) is quite trivial and i can get it.
    but i am wondering about (a) and (b) and (d), anyone know if (d) is proven using (b) and (c) or is there another method of doing it.
    I am having difficulty explaining (a), I think i understand why they are equal.. it is quite obvious, but i can't explain it well.
    and as for (b) i am also lost for words for the explanation

    any help would be greatly appreciated
  2. jcsd
  3. Mar 3, 2008 #2
    How about this explanation for (a): The minimum convex set of a convex set is itself and the result follows. For (b), pick two points in the union of Co(A) and Co(B) and show that they're also in Co(A U B).
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