Let $f=\sum_{n=1}^N k_n I_{E_n}$ where $E_n$ are Borel subsets of $\mathbb{R}^n$ and $k_n\in \mathbb{R}^m$ with non-negative entries, and let $\mu$ be a finite Borel measure on $\mathbb{R}^n$. What is the best rate of approximation, in $L^1(\mathbb{R}^n,\mathbb{R}^m)$ of $f$ by Lipschitz functions? That is, given any positive integer $n$ and some $L>0$, what order is: $$ \inf_{g \in Lip(\mathbb{R}^n,\mathbb{R}^m); Lip(g)\leq L}\,\int_{x\in \mathbb{R}^n} \|g(x)-f(x)\|d\mu(x)? $$

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with a given Lipschitz constant? $\endgroup$ – Pietro Majer Mar 16 at 18:151more comment