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

Dash <ddash.res@xxxxxxxxx> writes:

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."

In the usual statement of the theorem, the word "connected" isn't
included. (In fact, in the *usual* statement of the theorem, the word
"compact" isn't included either, though Caratheodory originally proved
it for the compact case only.) So, obviously, it's easy to come up
with simple examples in that case: let A consist of any two distinct
points.

For an example of a *connected* A where you still need d+1 points, I
don't believe you'll find one in R^1 or R^2, but you'll find it easy
to find one in R^3. Let A be the union of three line segments joined
at a single point (so their convex hull is a tetrahedron). Note that
no interior point can be expressed as a linear combination of only two
points of A.

--
Kevin Buhr <buhr+un@xxxxxxxxxxx>
.

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