Two questions:


1) (ALREADY ANSWERED) This is likely to be a very basic question for you folks. 


Carathéodory's theorem gives us an upper bound for the minimum number of convex hull vertices that can be used in a nonzero convex combination to yield an inner point of the convex hull (d+ 1 in $\mathbb{R}^d$). Is there a result which gives a *lower bound* on the *maximum* number of convex hull vertices that can be used in a nonzero convex combination to yield an inner point of the convex hull? (By an inner point I mean one that belongs to the convex hull but does not lie directly on the convex hull.)


I am primarily interested in whether any interior point of a convex hull can always be expressed as a nonzero combination of *all* convex hull vertices.


ANSWER: Any interior point of a convex hull can be expressed as a nontrivial convex combination of all hull vertices.


2) Would I be correct in saying that the convex hull of any set of points in a simplex is a Choquet simplex, which implies that in this case not only does such a nontrivial convex combination of convex hull vertices exist, but that the convex combination is unique?



Nonzero convex combinations of convex hull vertices to yield an inner point
This is indeed easy. Let $p$ be a point that you want to represent, $m$ the barycenter of all vertices and $\varepsilon>0$ so small that the point $q=(1+\varepsilon)p-\varepsilon m=p+\varepsilon(p-m)$ is still in the convex hull. Represent $q$ as a convex combination of some vertices, add $\varepsilon m$ with $m$ represented as the arithmetic mean of all vertices, and finally divide all the coefficients by $1+\varepsilon$.


