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I'm looking for a reference for the following theorem.

Theorem Let $X$ be a closed subset of $\mathbb{R}^N$, and let $a$ be a point of its convex hull $\operatorname{conv}(X)$. Then there exist affinely independent points $x_0, \ldots, x_n \in X$ such that:

  • $a \in \operatorname{conv}\{x_0, \ldots, x_n\}$, and
  • the only points of $X$ in $\operatorname{conv}\{x_0, \ldots, x_n\}$ are $x_0, \ldots, x_n$ themselves.

The hypothesis that $X$ is closed can't be dropped; it's easy to find examples in $\mathbb{R}^1$ showing this.

I'm not looking for a proof of this theorem, as I already have one (thanks to my student Adrián Doña Mateo). What I want is a reference. I imagine it's known, but I've been unable to find it, including in surveys of variants of Carathéodory's theorem.

Update Since no one answered here and we couldn't find it in the literature, we wrote up this theorem and a proof ourselves. It's Theorem 3.1 here:

Adrián Doña Mateo and Tom Leinster. Magnitude homology equivalence of Euclidean sets. arXiv:2406.11722, 2024.

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    $\begingroup$ for what it worth, here is a proof (maybe for some people it is easier to remember where they have seen a proof than a theorem): take a large closed ball $B$ such that $a\in \operatorname{conv}(X\cap B)$, then consider the simplex with vertices in $X\cap B$ of minimal dimension, and for this dimension of minimal volume, containing $a$ $\endgroup$
    – Fedor Petrov
    Commented Jan 13 at 19:01
  • $\begingroup$ Thanks, Fedor. This is indeed the proof I know, which my PhD student Adrián Doña Mateo came up with and told me. (That's not to say that it's not much older, of course; I expect it is.) $\endgroup$
    – Tom Leinster
    Commented Jan 13 at 20:45

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