Let $x\_1, \ldots, x\_n \in \mathbb{R}^d$ denote $n$ points in $d$-dimensional Euclidean space, and $w\_1, \ldots, w\_n \in \mathbb{R}\_{\geq 0}$ any non-negative weights.

$\arg\min\_{\mu \in \mathbb{R}^d} \sum\_{i=1}^n w\_i | x\_i-\mu| = median\{x\_1, \ldots, x\_n \}?$

I understand why the median of $\{x\_1, \ldots, x\_n \}$ minimizes the function without the weights. But I am not sure if the median is what minimizes the function with the weights, should I use Lagrange multiplier to solve this?