Let $\Omega\subset \mathbb{R}^{n}$ be a bounded domain that contain the origin and, is star-shaped with respect to the origin too. Is it true that, in spherical coordinates, $\partial \Omega$ is given by $r=r(\omega)$ so that $r$ is a function of the angle only?
2026-03-30 13:40:42.1774878042
Boundary of a star-shaped domain in spherical coordinates
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As N.Bach said, the answer is negative. For example, in $n=2$ take the unit disk minus the segment $[1/2, 1]$. This is a star shaped domain with respect to the origin but its boundary is not given as $r=r(\theta)$.
To have boundary representation $r=r(\theta)$ it suffices to assume that there exists a neighborhood of origin such that $\Omega$ is star-shaped with respect to every point in that neighborhood.