proof of Carathéodory’s theorem
The convex hull of consists precisely of the points that can be written as convex combination of finitely many number points in . Suppose that is a convex combination of points in , for some integer ,
where and . If , then it is already in the required form.
So, there are constants , not all equal to zero, such that
Let be a subset of indices defined as
Since , the subset is not empty. Define
Then we have
which is a convex combination with at least one zero coefficient. Therefore, we can assume that can be written as a convex combination of points in , whenever .
After repeating the above process several times, we can express as a convex combination of at most points in .
|Title||proof of Carathéodory’s theorem|
|Date of creation||2013-03-22 17:50:08|
|Last modified on||2013-03-22 17:50:08|
|Last modified by||kshum (5987)|