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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 $r\leq d$.

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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.

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