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, $x \mapsto \vert x \vert$: 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 $|x|$.

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

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.

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.

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, $x \mapsto \vert x \vert$: 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 $|x|$.