This is likely to be a very basic question for you folks.

Carathéodory's theorem gives us an upper bound for how many convex hull vertices are required to have nonzero weight in order for their 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 how many hull vertices with nonzero weight can be used in a convex combination of hull vertices to yield an inner point? (By an inner point I mean one that belongs to the convex hull but does not lie directly on the convex hull.)