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$A \subset [0,1]^n$ is a compact, convex subset of $\mathbb{R}^n.$ Take any subset (Not necessarily connected) $B \subseteq \partial A$, consisting of finite number of disjoint, closed subsets of the boundary of $A$. I want to show that every point in the convex hull of $B$ can be represented as the convex combination of at most two points of $B$.

This seems useful but apparently (according to comments) both the statement and the proof provided are inaccurate. This is the approach I have been taking so far. The part about induction on $n$ didn't occur to me but I tried to proceed using the supporting hyperplane theorem.

Any help is appreciated. Thanks in advance.

Canine360
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    This is not true if you take $B$ equal to the $2^n$ corners of the cube $A=[0,1]^n$. – daw Aug 02 '21 at 11:21
  • you might be interested in https://en.wikipedia.org/wiki/Carath%C3%A9odory%27s_theorem_(convex_hull) – daw Aug 02 '21 at 11:21
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    Thank you, this is a useful example which shows the way I have stated my problem is not correct. Actually the segments cannot be isolated points but let me figure out a correct way of putting exactly what I'm trying to say. I'm aware of the Caratheodory theorem but was hoping the reduction in the no. of points required was going to come from the fact that the set is part of the boundary of a convex set. It would be very helpful if you could point out what would be a correct "version" of the linked question and answer. That is almost exactly what I'm looking for. – Canine360 Aug 02 '21 at 11:34

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