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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