Compute a epsilon normal set

Question

Could you please help to me to compute the following $\epsilon$-normal set: Given $\epsilon>0$, how to compute the $\epsilon$-normal set of $C:=[2,\infty)\times \Bbb{R}$ at the point $(2,0)$. Thank you in advanced! An $\epsilon$-normal set of a convex set $C \subset \Bbb{R}^n$ at a point $z \in C$, denoted by $N_\epsilon(C,z)$, is the following set $\{u \in \Bbb{R}^n \mid \langle u, x-z \rangle \le \epsilon, \forall x \in C\}$

Answer 1

First find convex normal cone to $C$, at point $z=(2,0)$ which is

$$N(z;C)= \{u \in \Bbb{R}^n \mid \langle u, x-z \rangle \le 0, \forall x \in C\} = (- \infty, 0 ] \times \{0\} $$

Now observe that in finite dimension we have

$$ N_{\epsilon} (z ; C) = N(z;C) + \epsilon \Bbb B $$

Where $\Bbb B$ is closed Unit ball in $\Bbb{R^2}.$

$$ N_{\epsilon} (z ; C) = (- \infty, 0 ] \times [ - \epsilon , + \epsilon] \bigcup \epsilon \Bbb B $$