# Extension of Caratheodory’s Theorem

The Caratheodory’s Theorem says that any point in the convex hull of $$S \subset \mathbb{R}^d$$ can be represented by convex combination of no more than $$d+1$$ points in $$S$$. I'm wondering whether there are further results characterizing:

1. the subset of $$co(S)$$ where the above representation is unique. (I known that any point on the boundary of $$co(S)$$ yields a unique representation. I'm wondering whether there exists a larger subset.)

2. the subset of $$co(S)$$ in which each point can be represented by the combination of exactly $$r$$ points, where $$r\leq d$$.

If $$\operatorname{co}(S)$$ means the convex hull of $$S$$, then it's not true that any point on the boundary has a unique representation as a convex combination of points in $$S$$. For instance, with $$S$$ being the eight vertices of a 3-cube, none of the points in the relative interiors of the 6 2-faces (part of the boundary of $$\operatorname{co}(S)$$) are unique convex combinations; only points in the edges are.
And all the points in $$S$$ don't necessarily end up being vertices of $$\operatorname{co}(S)$$. For instance, if $$S$$ consists of the eight vertices of a 3-cube together with the 12 midpoints of its edges, then the points in the interior of the edges are no longer unique convex combinations; only the 8 vertices are unique convex combinations of points in $$S$$.
On the other hand, every point in a tetrahedron, or the $$d$$-simplex in any dimension $$d$$, is a unique convex combination of the vertices.
I think that in general the only points of $$\operatorname{co}(S)$$ which have a unique expression as a convex combination of points of $$S$$ are those contained in a simplicial face $$F$$ of $$\operatorname{co}(S)$$ such that $$F \cap S$$ contains only the vertices of $$F$$, and no redundant points.