Let $( \mathbb{R}^n, \| \cdot \|_P)$ be the $n$-dimensional Euclidean space equipped with $\ell_p$-norm $\| \cdot \|_p$ for some $p\in [1, + \infty]$. Let $A$ be a convex set in $\mathbb{R}^n$ and define \begin{align} A^{\epsilon} = \{ y \in \mathbb{R}^n \colon \exists x \in A~\text{such that}~\| x -y \| _{p} \leq \epsilon \}, \end{align} where $\epsilon >0$ is a real number. In general, what is the condition that there exists a $3$-order differentiable function $f \colon \mathbb{R}^n \rightarrow [0,1]$ such that $f$ is a good approximation of $\mathbb{1}_{A}$, which is the indicator function of set $A$.

In specific, it is ideal that $f(x) = 1$ when $x\in A$ and $f(x) = 0$ when $x\in \mathbb{R}^n \setminus A^{\epsilon}$. Moreover, the $\ell_q$-norm of the $i$-th order gradient of $f$ is proportional to $\epsilon^{-i}$ for $i = 1,2,3$. That is \begin{align} \| D^{(i)} f (x) \|_{q} \leq C_i \cdot \epsilon^{-i}, ~\text{for any}~~ i = 1,2,3. \end{align} Here the high-order gradients are taken as vectors, $1/q + 1/p = 1$, and $\| \cdot \|_q$ is the dual-norm of $\| \cdot \|_p$.

An inspiration of this problem is that for $p = q= 2$, the problem is solved for any convex sets. In addition, for rectangles of the form $\{ x \in \mathbb{R}^n \colon a_i \leq x_i\leq b_i, i =1, \ldots, n\}$, approximation under $\ell_{\infty}$ is also established. This approximation depends on the function $g(x） = 1/\rho \cdot \log [\sum_{i=1}^n \exp( \rho\cdot x_j)]$ which approximates the function that returns the maximum element of a vector in $\mathbb{R}%d$. See https://arxiv.org/abs/1412.3661v4. But general convex sets in $(\mathbb{R}^n, \| \cdot \|_{\infty})$ is not covered.

Moreover, for Banach spaces, similar results are also established for $\| \cdot \|_{B}$, which is the norm in the Banach space. However, such result depends on the differentiability of $\| \cdot \|_B$, which does not handle convex sets in $(\mathbb{R}^n, \| \cdot \|_{\infty})$.