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Given the $n$-dimensional triangulation $\mathbb{T}^n$ of a finite set of points $\{p_1,\ ...\ ,\ p_{n+k}\} \subset \mathbb{R}^n$, is it always possible to find $n+k$ positive real weights $\{\omega_1,\ ...\ , \omega_{n+k}\}$ so that all $n$-dimensional volumes of the simplices of $\mathbb{T}^n$ are equal when lifting their corner coordinates to homogenic coordinates via replacing $p_i$ with $(\omega_i p_i,\ \omega_i)\in\mathbb{R}^{n+1},\ i\in\{1,\ ...\ , n+k\}$?

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