A possible characterization for the median of a r.r.v.

Question

Let $X$ be a real random variable. It is clear that any median $m\in\mathbb{R}$ of $X$ satisfies that $$\text{E}[|X - m|] = \min_{x\in\mathbb{R}}\text{E}[|X - x|]$$.

My question is the following. If $b\in\mathbb{R}$ is a number such that $\text{E}[|X - b|] = \min_{x\in\mathbb{R}}\text{E}[|X - x|]$, is $x$ a median of $X$?

Thanks in advance.

Answer 1

Yes, if you define a median as a value $m$ where $\mathbb P(X<m) \le \frac12$ and $\mathbb P(X\le m) \ge \frac12$. Your $b$ will have this property.