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?