I've seen a lot of literature on Complex weighted projective spaces (sometimes defined in a flavour of algebraic geometry) which can be obtained as the orbit spaces of the action of $\mathbb{C}^*$ on $\mathbb{C}^{n+1}\setminus\{0\}$ defined by $$\lambda \cdot (z_0,\ldots, z_n):=(\lambda^{a_0}z_0,\ldots,\lambda^{a_n}z_n),$$ where $(a_0,\ldots,a_n)\subset\mathbb{N}^{n+1}$.
I imagine that we can define the same action of $\mathbb{R}^*$ on $\mathbb{R}^{n+1}\setminus\{0\}$ and look at the orbit space that what would be called a "real weighted projective space".
Is something known about these spaces, for example if they are compact and if they are orbifolds? Does anyone can give me a reference? I'll be very pleased, thanks!