# Approximate a Lipschitz function by an affine function [closed]

Suppose $$\mathcal{F}$$ is a Lipschitz function (Lipschitz constant $$L$$) space defined on $$A$$. $$A$$ is a bounded subset of $$\mathbb{R^n}$$. The $$\mathcal{G}$$ is an affine function space defined on $$A$$. The distance between $$\mathcal{F}$$ and $$\mathcal{G}$$ is defined as: $$\sup_{f\in \mathcal{F}} \inf_{g \in \mathcal G} ||f-g|| = \sup_{f\in \mathcal F} \inf_{g \in \mathcal G} \int_A (f(x) -g(x))^2 dx$$

Let me suppose $$\forall x \in A,~||x|\leq M$$.

## closed as unclear what you're asking by YCor, user44191, Piotr Hajlasz, Stefan Waldmann, Davide GiraudoMar 11 at 10:29

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• The statement is too general to be answered. If you have no further information about $A$ and $f$ you cannot compute the distance. – Piotr Hajlasz Mar 11 at 4:03