more symmetric definition again implies a set-valued median.

Let $\mathcal{X}$ be a finite multiset of real numbers. The median of $\mathcal{X}$ is defined as $$\begin{aligned}
\operatorname{med}(\mathcal{X}) &:= \phantom{{}\cap{}}
\left\{\mu\in\mathbb{R}~|~
\#(\mathcal{X}\cap(-\infty,\mu])\ge\tfrac12\#\mathcal{X}
\right\} 
\notag\\*&\phantom{{}:={}}
{{}\cap{}}
\left\{\mu\in\mathbb{R}~|~
\#(\mathcal{X}\cap[\mu,+\infty))\ge\tfrac12\#\mathcal{X}
\right\} \;.
\label{med-shalf}\end{aligned}$$

**The median as point of maximal half-line depth.**

The latter characterisation can be rephrased by defining the median as the set of points which split the data multiset as symmetric as possible. This characterisation of the median goes back to the work by Hotelling [@Hotelling-EJ29] and has been picked up by Tukey [@Tukey-icm74].

For a finite multiset $\mathcal{X}$ of real numbers, define the median of $\mathcal{X}$ as $$\begin{aligned}
\operatorname{med}(\mathcal{X}) &:= 
\mathop{\operatorname{argmax}}\limits_{\mu\in\mathbb{R}}
\min\{\#(\mathcal{X}\cap(-\infty,\mu]),\#(\mathcal{X}\cap[\mu,+\infty)\}
\;.
\label{med-hs}\end{aligned}$$ The count $\min\{\#(\mathcal{X}\cap(-\infty,\mu]),\#(\mathcal{X}\cap[\mu,+\infty)\}$ that indicates for any $\mu$ the number of data points that are contained at least in a half-line starting at $\mu$, is called *half-line depth* of $\mu$ w.r.t. $\mathcal{X}$.

Note that in a finite set (without multiplicities) the half-line depth of a data point is simply its rank or reverse rank. The extrema-stripping procedure from Definition \[def-med-extr\] then actually enumerates the data points in the order of increasing half-line depth; the half-line depth of each data point is the ordinal number of the step in 