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I have read about Radon's Theorem on Wikipedia. The theorem states that

Any set of $d + 2$ points in $\mathbb{R}^d$ can be partitioned into two sets whose convex hulls intersect.

I strongly suspect that the bound $d+2$ is even optimal, i.e. the statement above is not true for $\widetilde{d} < d+2$. This is easy to see in $\mathbb{R}$ and $\mathbb{R}^2$, but I do not see why this is also true for higher dimensions. Could you give me a hint?

3nondatur
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    Consider a set of basis vectors plus the origin. Any partition of this set in two subsets is separated by a linear inequality. – WimC Nov 12 '20 at 21:13
  • I am sorry but I can not follow you. We have to examine an ARBITRARY set of $d+1$ points in $\mathbb{R}^d$, so why do you consider such a set? – 3nondatur Nov 12 '20 at 22:08
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    The fact that there's a particular set of $d+1$ points can not be partitioned into two sets whose convex hulls intersect means that the statement with $d+2$ changed to $d+1$ is not true. – Robert Israel Nov 12 '20 at 22:22
  • @WimC: OK, I can intuitively see that this works, but is there a way to prove this in a formal way? Do you use the Separation Theorem here? How do we know that the convex hulls implied by the partition are compact and disjoint ? – 3nondatur Nov 13 '20 at 01:00

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