Convex sets - Convex Hull
Published by Angel on Wednesday, February 17, 2010 - 07:08:41 - Filed under Notes, Convex Sets
First post is on basic convex set theory, I liked this because it was simple, demonstration was not very hard either.
The convex hull for a set A ( Im using real A a subset of 
) is the minimal convex set that contains A.
We’ll now define:
Lemma:
Let 
with A a convex set and 
then 
Proof:
We’ll use induction over k.
If k=1 then x*1 is in A ( we choose x that way ).
Now we’ll asume that it works for every k.
Now, let 
where 
and 
We wanna see that x is in A.
Since then at least there is one 
, lets say its 
.
Let 
and
.
Then we know by our induction hypothesis that y is in A .
Now, since A is a convex set and 
then 
.
This shows that x is in A.
Definition:
A point 
its a convex combination of points 
if there is such that 
Theorem:
Let 
then c(A) ( The convex hull of the set A ) is the set of all convex combinations of all points in A.
Proof:
Let B the set of all convex combinations of points in A.
We now have to show that B=c(A) (convex hull of A )
1)
Let x be an element of B, then with 
and , since A is in c(A) then 
, then 
because c(A) is convex.
This shows that 
.
2)
Note that if B is a convex set then A is a subset of B,this is because if x is in A then there is a 
such that px will be on B. Then c(A) is a subset of B
We have to show that B is convex.
Let x and y be in B
Then and 
where 
and and 
.
Let 
with 
such that 
Then 
, it is clear now that 
This shows that 
, which means B is convex.
Then B=c(A).
——-
‘Till next time :)
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