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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.

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?

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.)

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.

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.)

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.)