# Tag: absolute-value

## real analysis – Reverse Triangle Inequality Proof

The Question : 106 people think this question is useful I’ve seen the full proof of the Triangle Inequality \begin{equation*} |x+y|\le|x|+|y|. \end{equation*} However, I haven’t seen the proof of the reverse triangle inequality: \begin{equation*} ||x|-|y||\le|x-y|. \end{equation*} Would you please prove this using only the Triangle Inequality above? Thank you very much. The Question Comments :

## optimization – The median minimizes the sum of absolute deviations (the ${\ell}_{1}$ norm)

The Question : 119 people think this question is useful Suppose we have a set $S$ of real numbers. Show that $$\sum_{s\in S}|s-x|$$ is minimal if $x$ is equal to the median. This is a sample exam question of one of the exams that I need to take and I don’t know how to