What kind of hyper-surfaces, functions I am dealing with here?
A smooth hyper-surface $S^{n-1}$ in $R^{n}$ is defined by an equation $(x\frac{\partial f(x)}{\partial x})=\rho>0$ with a smooth positive definite (non-homogeneous) function $f(x)>0$ for $\forall x\in R^{n}\setminus0$ and an assumption that a regular (finite) solution of the above equation exist for all radial directions of $x$.
Looks like that it is homeomorphic to a (n-1)-sphere in $R^{n}$ with $0$ as its volume internal point. It seems to be kind of convex, but not in terms of any strict notion of convexity I am aware of. What are those types of surfaces in general? Or these types of functions?
Let us now include the possibility of (limit to) $\rho=0$. How to make sure that the function $f(x)$ is such, that equation $(x\frac{\partial f(x)}{\partial x})=0$ has the only solution $x=0$? Will positive Hessian suffice? Positive semi-definite in a vicinity of $0$ only? Will it be necessary, or sufficient, or both? Are there any theorems on this?
I'll be really grateful for a help in clarifying the concepts and for useful specific references.
Thank you.