# Convex hull and geometric median

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 $$q>0$$?