Let $S$ be a set of points in $\mathbf{R^d}$ and let $m$ be the median of this set of points, i.e. $\sum_{x \in S} || x - y||$ is minimized when we have $y=m$. Now let $z$ be an arbitrary point in $\mathbf{R^d}$. Is the following true?
$$\sum_{x \in S} ||x-z|| \leq K \cdot (\sum_{x \in S} ||x-m|| + |S| \cdot ||m-z||)$$ where $K$ is a constant.
We know that if we replace all the $||a-b||$ expressions by $||a-b||^2$, and we consider the mean instead of the median(i.e. replace $m$ with $\mu$), then actually we will have equality and $K=1$. So I want to know if similar things hold for the median.
I was thinking maybe the following is true. Is it?
$$\sum_{x \in S} ||x-z|| \leq \cdot (\sum_{x \in S} ||x-m|| + 4|S| \cdot ||m-z||)$$