We will be smoothing various functions later on in this section so let us recall the standard regularization techniques. Given a convex bounded domain $P\subset\mathbb{R}^{n}$ with piece-wise smooth boundary let $\rho$ be a mollifier whose support is $P$, i.e. a positive $\mathbb{C}^{\infty}(\mathbb{R}^{n})$ function vanishing exactly outside $P$ and such that $\int_{\mathbb{R}^{n}}\rho dx=1$. For instance, if $P$ is a polytope in $\mathbb{R}^{n}$ given by a collection of inequalities $\{\langle v_{i},x\rangle+\lambda_{i}\leq 0\}$ one can take the usual bell-shaped function

$$\rho=c\prod_{i}\rho_{i},\quad\text{ where }\rho_{i}(x)=\begin{cases}e^{\frac{1}{\langle v_{i},x\rangle+\lambda_{i}}},&\quad\langle v_{i},x\rangle+\lambda_{i}\leq 0\\ 0,&\quad\langle v_{i},x\rangle+\lambda_{i}\geq 0\end{cases},$$

and the constant $c$ is determined from the normalization. Set $\rho_{h}:=\frac{1}{h^{n}}\rho(\frac{x}{h})$. For any locally integrable function $u\in L^{1}_{loc}(\mathbb{R}^{n})$ we can apply the standard regularization procedure by taking the convolution with $\rho_{h}$:

$$u_{h}(x):=\rho_{h}\ast u=\int_{\mathbb{R}^{n}}\rho_{h}(x-y)u(y)\,dy.$$

Then, if $u\in C^{p}(\mathbb{R}^{n})$, the functions $u_{h}$ are $\mathbb{C}^{\infty}$ and approaching $u$ as $h\to 0$ in the $C^{p}$-norm uniformly on any compact in $\mathbb{R}^{n}$.

Below we list some elementary properties of $u_{h}$ which will be useful later.

One can also apply the regularization procedure by taking the convolution with the mollifier parameter depending (smoothly) on the point $x\in\mathbb{R}^{d}$. That is $u_{h}(x):=\int_{\mathbb{R}^{n}}\rho_{h(x)}(x-y)u(y)\,dy$. Or, even, more generally the entire shape of the support of the mollifier can smoothly depend on the center of convolution. We have used this technique of varying support to prove the existence of bi-PIKAS in Proposition 2.1.