Re: Example of a set requiring d+1 points (Caratheodory theorem)

Dash wrote:

Hi,
I have a trivial doubt about a form of Caratheodory theorem which has
applications in Information Theory. The statement of the theorem is,
"Any point in the convex closure of a connected compact set A in a d
dimension Euclidean space can be represented as a convex combination
of d+1 or fewer points in the original set A."

My problem is that I am unable to find any example where d+1 points
are required. (It is easy to find examples where only d points
suffice). In addition, is there an example in R^n?

I wasn't familiar with the theorem, but looking at

http://en.wikipedia.org/wiki/Carathéodory's_theorem_(convex_hull)

I don't see any hint of A having to be connected. If you drop that
requirement, any set of two points in R^1 is an example.
Maybe that explains it.

--
Niels Diepeveen
.

Relevant Pages