can again be rewritten in a half-line depth form, where the continuous half-line depth is determined by integrals.

For a normalised regular density $\gamma$, its median is defined as $$\begin{aligned}
\operatorname{med}(\gamma) &:=
\mathop{\operatorname{argmax}}\limits_{\mu\in\mathbb{R}}
\min\left\{
\int\nolimits_{-\infty}^{\mu}\gamma(x)\,\mathrm{d}x,
\int\nolimits_{\mu}^{+\infty}\gamma(x)\,\mathrm{d}x
\right\}
\;.
\label{medd-hs}\end{aligned}$$

**Continuous extrema stripping.**

Even an extrema-stripping formulation of the continuous median is possible. To this end, a the stripping procedure must be stated as a time-continuous process. We will give this definition for simplicity in the case of a density with a connected compact support set.

Let $\gamma$ be a normalised regular density which is zero outside some interval $[a_0,b_0]$ and nonzero in its interior $(a_0,b_0)$. Consider then the ordinary differential equations $$\begin{aligned}
a'(t)&=\bigl(\gamma(a(t))\bigr)^{-1}\;,&
b'(t)&=-\bigl(\gamma(b(t))\bigr)^{-1}
\label{medd-exstrip-ode}\end{aligned}$$ with the initial conditions $$\begin{aligned}
a(0)&=a_0\;, & b(0)&=b_0\;.\end{aligned}$$ This initial boundary value problem yields a monotonically increasing $a:[0,T]\to\mathbb{R}$ and monotonically decreasing $b:[0,T]\to\mathbb{R}$ on an interval $[0,T]$ where $T$ is chosen such that $a(T)=b(T)$, and the median of $\gamma$ is defined as $$\begin{aligned}
\operatorname{med}(\gamma) &:= a(T)\;.\end{aligned}$$ For each $t\in[0,T]$, we call $t$ the continuous half-line depth of $a(t)$ and $b(t)$.

**Minimisation property.**

Provided that $\gamma$ is compactly supported, a reformulation using a minimisation property is again possible. This median is called *geometric median* in [@Haldane-Biomet48].

Let 