Given a curve $f(x)$ (for $x \in [0,1]$), and a line $y=a$, let $U$ be the total area below $f$ and above $a$, and let $L$ be the total area above $f$ and below $a$. If $L=U$, this means that $a =\int_0^1 f(x) dx$.
Or if $f(x)$ is a sequence of $n$ numbers, this means $a$ is their mean. Equivalently, $U = \int_0^1 \max(f(x)-a,0) \; dx$, $L = \int_0^1 \max(a-f(x),0) \; dx$.
What I want instead is the value $a$ for which $c \cdot L = U$, for some constant $c>0$.
This can be computed approximately by binary search or exactly with a more cumbersome algorithm, but: is there a closed-form expression for this value? And: has this value been studied?