What are constraints on the "looks" of a function $f:\mathbb{R}^2\rightarrow \mathbb{R}$, if $f$ satisfies $f(x_1+x_2)\leq f(x_1) + f(x_2), \ \quad x_1,x_2\in \mathbb{R}^2 \quad \quad (1)$i.e. the triangle inequality ?
An answer should be something analoguous to: if $f$ satisfies $f(\lambda x_1)=|\lambda| f(x_1), x_1\in \mathbb{R}^2, \lambda \in \mathbb{R} $, then $f$ is a function symmetric with respect to the planes defined be $y=0$ and $x=0$ (if $x$ and $y$ denote the first and second component of some vector $x_1$), such that on every line through the origin, $f$ is a linear function. If we add continuity, then $f$ can be thought of to be glued together by planes and sections of this function (which illstrates one extreme: Total roundness - vs. the function $(x,y)\mapsto y $ or $x$ which, if refrain from considering trivial functions, illustrates the other extreme of being totally not-round).
So I guess I'm actually looking for properties describing the solution set of the "functional inequality" of $(1)$.
By the way, this question arose when I was trying to visualize how real functions in the plane look, if they satisfy the norm axioms. So even if you can't answer the above (more abstract) questions, maybe you can contribute to this.
EDIT (In response to M. Sleziaks comment) I would also welcome:
a) every answer that summarizes only the geometric aspects of subadditive functions in the book Martin Sleziaks named (since I'm rather short of time to dig through the whole chaprter that also covers algebraic and topologic/differenitiability results), if someone maybe already has read it.
b) geometric properties of real functions that satisfy all norm axioms (this may make the treatment easier since already the geometric results I described, that follow only from the positive homogeneity axiom, seem to me to be rather restricting)