Let $O\subset\mathbb{R}^n$ be a open set which is star shaped with respect to the origin. How does one prove that there exists an increasing sequence of star shaped (w.r.t the origin) domains $O\_i$ such that : 


* $O\_i$ has a smooth boundary
* $O\_i\subset O$
* $\lambda (O\backslash O\_i)\to 0$ when $i\to\infty$, or, even better for my purpose, $O\backslash O\_i$ is contained in an $\varepsilon\_i$-neighborhood of $\partial O$.


For my purpose, it wouldn't hurt to assume that $O$ is bounded if that helps. This kind of approximation is handy for estimating the distributional laplacian of the distance function on a Riemannian manifold. 


That seems like a pretty standard fact but I haven't been able to locate or derive a clean proof of this. I apologize if the question is not up to the standards of MO.



Can one smooth open star shaped domains from the inside by star shaped domains?
This is not really an answer, but an equivalent and hopefully easier reformulation of the question that does not fit in one comment.


You can describe the starshaped domain $O$ by a function $\rho\colon S^{n-1}\to(0,\infty]$ such that $$O\setminus\{0\}=\{\,x\in\mathbb R^n\setminus\{0\}\mid |x|<\rho(x/|x|)\,\}\;.$$ Then $O$ is open if and only if $\rho$ is lower smicontinuous. You want to approximate $\rho$ by smooth functions $\rho\_i$ from below. This turns out to be impossible (both with respect to Lebesgue measure and with respect to distance) if $\rho$ jumps from $\infty$ down to a finite number.


If $\rho$ is bounded by $C$, the "$\varepsilon\_i$" condition is equivalent to the following: all $\rho\_i<\rho\_{i+1}<\rho$ for all $i$, and for each $\varepsilon>0$, there exists an $i$ such that for each $p\in S^{n-1}$, there is $q\in S^{n-1}$ with $|q-p|<\varepsilon$ such that $\rho\_i(p)>\rho(q)-\varepsilon$. The idea is that there exists $r\ge \rho(q)$ such that $|rq-\rho\_i(p)\,p|<(1+C)\varepsilon$, and there is a boundary point of $O$ somewhere on the line segment from $\rho\_i(p)\,p$ to $rq$.
I am sure one can show that smooth approximations exist that satisfy this.


If $\rho$ is unbounded, one has to demand instead that for each $\varepsilon>0$, there exists an $i$ such that for each $p\in S^{n-1}$, there is $q\in S^{n-1}$ with $|q-p|<\varepsilon/\rho\_i(p)$ such that $\rho\_i(p)>\rho(q)-\varepsilon$. Just adapt the argument above.


