The question I'm working on, (a version of) which I have been posting in a piecemeal manner over the last few days because I thought they would be of more general interest that way, is really the following:
Take a compact convex set $A \subset \mathbb{R}^n$. Take its boundary $\partial A$. Take its intersection with the $n$-dimensional probability simplex $\Delta_n=\{x \in \mathbb{R}^n: x\geq 0, \sum\limits^n_ix_i\leq 1\}$, i.e. take $\partial A \cap \Delta_n = : B$ (say). Assuming $B \neq \emptyset$ (not that this matters), can every point in the convex hull of $B$ be represented as a convex combination of at most two points in $B$?
As explained in a (now deleted) answer to this question of mine (and as is kind of obvious), the answer is yes if $\partial A \cap \Delta_n = \partial A$ or $\partial \Delta_n$. What about in general?
I'm totally open to the possibility that this may not be true, and a counterexample would also be most appreciated.
Thanks a lot for your help.
Best Answer
Both the simplex and A are convex so their intersection will be convex as well. If $A\subset \Delta_n$ then it is trivially true that any point in B is a convex combination of two points in $\partial A$.
If not, then there are some points that are in B but not A. These will be along a line joining two points in $\partial A$.
Therefore, either way—