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Let $X\subseteq\mathbb R^d$ ($d\geq 1$) and let $\Delta_{(X)}$ be the collection of all families $(w_x)_{x\in X}$ of nonnegative numbers, such that $\{x\in X:w_x\neq 0\}$ is finite, and that sum to one (i.e., the collection of all finitely supported probability measures on $X$). A geometric median of $X$ with weights $w\in\Delta_{(X)}$ is any minimiser of $\sum_{x\in X}w_x\|t-x\|_2$, over $t\in\mathbb R^d$.

Is it true that the collection of all geometric medians of $X$, with respect to all possible weights $w\in\Delta_{(X)}$, coincides with the convex hull of $X$?

And is it true if I change $\|\cdot\|_2$ with some other functions, such as $\|\cdot\|_p^q$, with $p\geq 1$ and $q>0$?

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