Let $f:\mathbb{R}^n\rightarrow\mathbb{R}$ be a nonlinear convex function, and let $x\in\mathbb{R}^n$ be an arbitrary vector. Define the set $\Omega_f(x)$ as
$\Omega_f(x) \triangleq \{z\in\mathbb{R}^n: f(x)+f(z) = f(x+z)\}$
If $f(x) = \|x\|_2^2$, then $\Omega_f(x)$ is just the set of vectors orthogonal to $x$. Are there other functions $f$ for which $\Omega_f(x)$ has a nontrivial closed-form solution? For instance, if $f(x) = \sum_{i=1}^n \mathrm{log}(x_i)$, then $\Omega_f(x)$ is the set of solutions to $\prod_{i=1}^nx_iz_i = \prod_{i=1}^n(x_i+z_i)$. Does this have a closed form solution?