Variational methods - properties of convex hull

  • #1
162
0
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
 

Answers and Replies

  • #2
1,357
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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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