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.

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