Suppose $f: A \rightarrow \mathbb{R}$ is a Lipschitz function (Lipschitz constant $L$), and $A$ is a bounded subset of $\mathbb{R}^n$ (i.e., for any $x \in A, ||x||<M$).
If I am going to use an affine function to approximate this Lipschitz function, what is the $\ell 2$ norm of their difference? That is, what is
$$\sup_{f} \inf_{g} ||f-g|| = \sup_{f} \inf_{g} \int_A (f(x) -g(x))^2 dx$$
$f$ is defined above, and $g$ is an affine function defined on $A$.