Suppose $\Omega \subset \Bbb R^n$ be a domain such that $|\Omega|<\infty$, $f\in L^1(\Omega)$. Let $Y= \text{span}\{g_1,\dots, g_k\}$.

Is there a characterization of the set of

projectionsof $f$ onto $Y$, i.e. the set $M$ of all $g\in Y$ such that $$ \int_\Omega |f(x) - g(x)|dx = \inf_{h\in Y} \int_\Omega |f(x) - h(x)|dx, $$ or, equivalently, $M = \text{argmin} \{ \int_\Omega |f(x) - g(x)|dx : g \in Y\}$?

Similar problem in $L^2$ (minimizing $\int_\Omega |f(x) - g(x)|^2 dx$) is very easy to solve since we have inner product there. However, as $L^1$ norm is not strictly convex, our set $M$ need to contain a unique element in general.

The solution to the $1$-dimensional case $Y = \text{span}\{1\}$ is well known, i.e. $g=c$ where $c\in \Bbb R$ is any median of $f$. Is there a complete treatment of this kind of problem anywhere?

Thank you in advance.