Median minimizes expected absolute loss ($\arg {\min }_a\mathbb{E}|X-a|$)

For a finite first absolute moment, the minimizers of expected absolute loss are the medians. With a density, differentiating the loss gives $2F(a)-1$, so every point where the cumulative distribution function equals one half minimizes it. A density gap may give a whole minimizing interval. In contrast, finite squared loss is uniquely minimized by the expectation.