Equivalence of convex homogeneous functions and pointwise maximum of linear functions I've seen the following question and solution

Question When is the epigraph of a function
  a convex cone?
Solution If the function is convex and positively homogeneous $(f(\alpha x) = \alpha f(x)$ for $\alpha \geq 0$).

The solution I came up with was that the function must be the pointwise maximum
of a finite number of linear functions, i.e.
$$
f(x) = max(f_1(x), ..., f_n(x))
$$
where $f_1$, ...,  $f_n$ are linear.
This lead me to wonder, is the set of functions defined by the pointwise maximum of a finite number of linear functions equivalent to the set of convex positively homogeneous functions? If not, what is wrong with my solution above? Does this equivalence hold for a countably infinite set of linear functions?
Edit
Since this question is derived from epigraphs, I am interested in functions $f: \mathbb{R}^M \rightarrow \mathbb{R}$ and similarly for 
$f_1, ...,  f_n$.
 A: your claim is not true.  Take $f(x,y)= \sqrt{x^2 + y^2}.$ 
$f$ is convex and homogeneous function, but not convex piece-wise function. 
But any convex homogeneous  function can be expressed as the pointwise supremum of  linear function. See Variational analysis by Rockafellar, Theorem 8.13
It is not difficult to adopt the proof for homogenous convex function. Note that for convex homogenous function you may assume those affine functions supporting $f$ pass through origin, this makes affine functions indeed linear! Secondly you can pick just at most countable many of them since $f$ is continuous and $Q^n$ is dense in $R^n.$  Therefore any convex homogenous function, $f: R^n \rightarrow R$ can be equivalently written in the following form:
$$ f(x) = \sup_{n \in N} \{f_n(x)\}  $$ where $f_n$ is linear for all $n\in N.$
A: You see in a moment that, in $\mathbb{R}^n$ with $n\geq 2$, your result cannot be correct.
Consider, for example, the convex and positively homogeneous function $f(x) = |x|$.
This function is of class $C^1(\mathbb{R}^n\setminus\{0\})$, whereas the pointwise maximum of a finite family of linear functions cannot be differentiable on that set. (Roughly speaking, look at the graphs of the $f_i$: at "lines of junction" the function is not differentiable.)
