These are the definitions I'm using (cf Rockaffeller):
- normal cone to a convex set $C$:
$$\mathcal{N}_C(x)=\{d\ | <d,y-x>\leq 0,\ \forall y\in C\}$$ - tangent cone to a convex set $C$:
$$\mathcal{T}_C(x)=\{u\ | <d,u>\leq 0,\ \forall d\in\mathcal{N}_C(x)\}$$
Let's take $C=\mathbb{R}_+$ (which is convex):
if I am not mistaken, $$\begin{equation}
\mathcal{N}_{\mathbb{R}_+}(x)=\begin{cases} \{0\} & \text{if $x>0$} \\
\mathbb{R}_- & \text{if $x=0$}
\end{cases} \quad\text{ so }\quad\mathcal{T}_{\mathbb{R}^+}(x)=\begin{cases} \mathbb{R} & \text{if $x>0$} \\
\{0\} & \text{if $x=0$}\end{cases}
\end{equation}$$
the normal cone of which I found to be:
$$\mathcal{N}_{\mathcal T_{\mathbb{R}_+}(x)}(y)=\begin{cases}
\{0\}&\text{if $x>0$} \\
\begin{cases} \mathbb{R}^+ & \text{if $y>0$} \\
\mathbb{R} & \text{if $y=0$} \\
\mathbb{R}_-&\text{if $y<0$}
\end{cases}&\text{if $x=0$}
\end{cases}$$
Did I make a mistake in the calculation? For some reason, I was excepting not to see $\mathbb{R}$ (maybe $\{0\}$) in the latter expression.
Best Answer
I indeed made a mistake; I give the actual results after verification, in case it helps someone (feel free to delete the question though): $$\begin{equation} \mathcal{N}_{\mathbb{R}_+}(x)=\begin{cases} \{0\} & \text{if $x>0$} \\ \mathbb{R}_- & \text{if $x=0$} \end{cases} \quad\text{ so }\quad\mathcal{T}_{\mathbb{R}^+}(x)=\begin{cases} \mathbb{R} & \text{if $x>0$} \\ \mathbb{R}_+ & \text{if $x=0$}\end{cases} \end{equation}$$
the normal cone of which I found to be: $$\mathcal{N}_{\mathcal T_{\mathbb{R}_+}(x)}(y)=\begin{cases} \{0\}&\text{if $x>0$} \\ \begin{cases} \{0\} & \text{if $y>0$} \\ \mathbb{R}_- & \text{if $y=0$} \\ \end{cases}&\text{if $x=0$} \end{cases}$$