I am reading a preliminary version of a paper which focuses on some minimization problems connected to a class of integral functionals. Reading the assumptions of one of the theorems I cannot convince myself that a "concrete" example exists (beside the "trivial" function, $x \mapsto \alpha \vert x \vert$ for some $\alpha >0$: the authors want to extend some previous result on this case to a more general framework). 


**Question.** Does there exist a function $f \colon \mathbb R^N \to [0,+\infty)$ such that 


1. $f$ is convex;
2. $f(\lambda x) = \lambda f(x)$ for any $\lambda >0$ and $\forall x \in \mathbb R^N$;
3. there are $a>0, b \ge 0$ and $\gamma \in \mathbb R^N$ such that 
$$
a|x| \le f(x) + \langle \gamma, x \rangle + b
$$
for any $x \in \mathbb R^N$?


I have some problems in finding a function satisfying the three points... It does not have to be smooth, still I do not see an example beside $\alpha |x|$, $\alpha >0$. 



Example of convex functions fulfilling a (strange) lower bound
Let $G$ be the set of functions $g\colon\mathbb R\to\mathbb R$ such that for some strictly positive real $a$ and $b$ and all real $x$ we have $g(x)=-ax$ if $x\le0$ and $g(x)=bx$ if $x\ge0$. Let $l\_1,\dots,l\_N$ be any linearly independent linear functionals on $\mathbb R^N$. Then any function $f$ on $\mathbb R^N$ of the form 
\begin{equation\*}
 f=\sum\_1^N g\_i\circ l\_i
\end{equation\*}
with the $g\_i$'s in $G$ will be nonnegative and satisfy your conditions 1, 2, 3. 
More generally, we can take 
\begin{equation\*}
 f=\sum\_1^n g\_i\circ l\_i \tag{0}
\end{equation\*}
with the $g\_i$'s in $G$, where $l\_1,\dots,l\_n$ are any linear functionals on $\mathbb R^N$ spanning $(\mathbb R^N)^\*$. 


**Added in response to a comment by the OP:** Here are details on why the so-constructed $f$ will satisfy condition 3. Let 
\begin{equation}
 c:=\inf\_{x\ne0}\frac{f(x)}{|x|}=\min\_{|x|=1}f(x), \tag{1}
\end{equation}
since $f$ is positive homogeneous and continuous. 
Suppose that $f(x)=0$ for some $x\in\mathbb R^N$. Since $g\_i\ge0$, (0) implies $g\_i(l\_i(x))=0$ for all $i$. Since $g\_i(u)=0\implies u=0$, we have $l\_i(x)=0$ for all $i$, and hence $l(x)=0$ for all $l\in(\mathbb R^N)^\*$, since the $l\_i$'s span $(\mathbb R^N)^\*$. So, $f(x)=0$ implies $x=0$. So, (1) implies $c>0$ and $c|x|\le f(x)$ for all $x\in\mathbb R^N$, so that condition 3 indeed holds. 


