# The geometry of functions $\mathbb{R}^2\rightarrow \mathbb{R}$ that satisfy the norm axioms

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)

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I am not sure to which extent this will be useful for you, but such functions are called subadditive. Chapter 16 of the book Kuczma: An Introduction to the Theory of Functional Equations and Inequalities is devoted to subadditive functions. The subadditive functions from $\mathbb R^n$ to $\mathbb R$ are studied in this book. –  Martin Sleziak May 29 '12 at 17:05
@MartinSleziak Ok, I glanced in the book; it seems the theory of subadditive functions is richer than I thought... –  temo May 29 '12 at 17:51
But since the book is big enough and I don't plan to read through a whole book to get one answer, I'm going to offer a bounty... –  temo Sep 1 '12 at 16:45
what do you mean "the looks" –  Squirtle Sep 4 '12 at 3:47
@dustanalysis with "constraints on the 'looks'" I mean: What are the shapes a function satisfying the triangle inequality absolutely cannot have. I have already given examples of what I mean in the case of function satisfying the axiom of homogeneity: These have to have a certain symmetric shape as explained in my question; no other shape is possible. –  temo Sep 5 '12 at 10:14