# Existence of a convex hull of few points in R^n ?

• Jan 12th 2008, 02:56 AM
sirandreus
Existence of a convex hull of few points in R^n ?
Hi,

I have a set of vectors V {v1,...,vk} where every element of V is from R^n. Please consider that k could be smaller than n.

Does the convex hull C exist for the vectors in V ? so that v1,...,vk are elements of C

If I write a vector v' as a convex combination of the vectors V, will v' be an element of C as well ?

v'=c1*v1+...+ck*vk
(where c1+...+ck = 1 and c1,..,ck positive)

from my point of view it should work.

consider for instance R^3.
- two random vectors v1 and v2, form a convex set as a line which connects v1 and v2. Thus any vector on the line can be expressed as a convex combination of v1 and v2
- three random vectors v1, v2, v3 form a triangle. And any convex combination will be inside the triangle
- vector counts higher then the dimensions, would form a polyhedron and there could be interior vector as well (vectors inside the convex hull).

It is clear to me for trivial example in low dimensional spaces. However Im not sure if this is valid for any dimensionality as well.

kind regards,
Andreas
• Jan 18th 2008, 05:26 AM
Rebesques
Quote:

Does the convex hull C exist for the vectors in V ? so that v1,...,vk are elements of C

Yes, ofcourse it does.

Quote:

If I write a vector v' as a convex combination of the vectors V, will v' be an element of C as well ?

Most certainly.

Quote:

- two random vectors v1 and v2, form a convex set as a line which connects v1 and v2.
The convex hull is the whole triangle.

Quote:

- three random vectors v1, v2, v3 form a
Random tetrahedron, which (sides and interior included) forms the convex hull.

But note that in more general spaces, the convex hull does not only consist of the set of convex combinations. Remember the Krein-Milman theorem.
• Jan 18th 2008, 06:17 AM
sirandreus
Hi Rebesques,