Given an nondegerate ellipsoid $E$ in $\mathbb{R}^d$, described as $E = \{x\in\mathbb{R}^d: (x-x\_0)^TQ\_0(x-x\_0)\leq 1\}$ and let $\chi\_E$ be the characteristic function supported on $E$. I am thinking about how to construct a family $\{f\_n:\mathbb{R}^d\to\mathbb{R}\}\_{n= 1}^\infty$, where each $f\_n$ can be written as a smooth expression in terms of $Q\_0, x\_0$ and $n$, such that, as $n\to\infty$, $f\_n\to\chi\_E$ in $L^p$, for all $p\geq 1$. I have googled related key words but found nothing. Any advice or related reference would be appreciated. Thanks. 



Family of funcitons that approximates uniform density on an ellipsoid
There are many good ways to accomplish this. One is by using the logistic function $L$, defined by the formula 
$$L(u):=\frac1{1+e^{-u}}.$$
Then 
$$g\_n(u):=L(n(1-u))
\left\{
\begin{aligned}
\uparrow1&\text{ if }0\le u<1,\\
\downarrow0&\text{ if }u>1 
\end{aligned}
\right.
$$
as $n\to\infty$. 
So, assuming that $Q\_0$ is positive definite, letting 
$$f\_n(x):=g\_n((x-x\_0)^TQ\_0(x-x\_0)) 
$$
for all $x\in\mathbb R^n$, and using (say) the dominated convergence, we conclude that $f\_n$ converges to the characteristic function of the set $E$ in $L^p$ for each $p\ge1$, as desired. 


