Consider a smooth function $f(x) \colon \mathbb{R}^2_+ \to \mathbb{R}_+$ such that $f$ is concave and positively homogeneous of order one. Consider a linear transform $P$ given by matrix $$ P = \begin{pmatrix} p_1 & 0 \\\ 0 & p_2 \end{pmatrix}, $$ where $p_1, p_2 > 0$ and $p_1 \neq p_2$ if $p_1 = 1$ of $p_2 = 1$. I have a hypothesis that the system of equations $$ \left \{ \begin{array}{rcl} f(x) & = & 1 \\ f(Px) & = & 1 \end{array} \right. $$ can't have more that one solution under some general conditions on function $f(x)$. Geometrically it means that curves $f(x) = 1$ and $f(Px) = 1$ can't have more than one common point (such curves are called pseudolines). Help me please with any idea to find such conditions on function $f(x)$. I tried to use some version of the global inverse function theorem, but unsuccessfully.
