Example of convex functions fulfilling a (strange) lower bound

Question

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 $\alpha |x|$, $\alpha >0$.

Answer 1

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.

Answer 2

Another construction that works is the following: Take any convex $A\subset\mathbb{R}^n$ which contains a neighborhood of the origin. Then the associated Minkowski functional $$ \sigma_A(x) = \inf\{\lambda>0\mid x\in\lambda A\} $$ has the desired properties.