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

TheoremLet $X$ 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.