Re: (non)complete graphs of polytopes and (non)unique decomposition as a convex combination of vertices

* Erik Quaeghebeur <equaeghe@xxxxxxxxxxxxxx>:

I think (but am not sure) that when one has a polytope with a complete
graph as adjacency graph (of its vertices), any point of the polytope
can be written as a unique convex combination of these vertices _and_
that the converse also holds: a polytope with a noncomplete graph as
adjacency > graph does not have this property,


On Fri, 6 Jul 2007, Michael Slone wrote:

Yes, you are correct on both counts. Radon's theorem says that
the vertex set of a polytope of dimension d with more than d+1
points can be partitioned into two blocks whose convex hulls
intersect, implying your second claim above.

Thanks, Micheal, mentioning Radon's theorem was useful (I didn't know it
before). The one thing that strictly speaking, I still have to solve, is
going from the (number of edges is the) adjacency graph to the dimension
of the corresponding polytope (usually much smaller than the space it's
embedded in, in my case).

Any suggestions welcome,

Erik
.

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