Let $p \in \mathbb{N}^{\star}$. We assume that $f, g_1, \ldots, g_p$ are functions from $\mathbb{R}^n$ to $\mathbb{R}$ differentiable on $\mathbb{R}^n$, and that $$K = \left\{ x \in \mathbb{R}^n, g_1(x) \leqslant 0, \ldots, g_p(x) \leqslant 0 \right\}$$ is non-empty. For all $x \in K$, we denote $I_x = \left\{ i \in \llbracket 1, p \rrbracket, g_i(x) = 0 \right\}$. We consider $x^{\star} \in K$ and we make the following hypothesis: $$(H) \quad \text{there exists } v \in \mathbb{R}^n \text{ such that for all } i \in I_{x^{\star}}, \langle \nabla g_i(x^{\star}), v \rangle < 0.$$ Show that $\mathcal{A}_K(x^{\star}) = \left\{ h \in \mathbb{R}^n, \forall i \in I_{x^{\star}}, \langle \nabla g_i(x^{\star}), h \rangle \leqslant 0 \right\}$.
Let $p \in \mathbb{N}^{\star}$. We assume that $f, g_1, \ldots, g_p$ are functions from $\mathbb{R}^n$ to $\mathbb{R}$ differentiable on $\mathbb{R}^n$, and that
$$K = \left\{ x \in \mathbb{R}^n, g_1(x) \leqslant 0, \ldots, g_p(x) \leqslant 0 \right\}$$
is non-empty. For all $x \in K$, we denote $I_x = \left\{ i \in \llbracket 1, p \rrbracket, g_i(x) = 0 \right\}$.\\
We consider $x^{\star} \in K$ and we make the following hypothesis:
$$(H) \quad \text{there exists } v \in \mathbb{R}^n \text{ such that for all } i \in I_{x^{\star}}, \langle \nabla g_i(x^{\star}), v \rangle < 0.$$
Show that $\mathcal{A}_K(x^{\star}) = \left\{ h \in \mathbb{R}^n, \forall i \in I_{x^{\star}}, \langle \nabla g_i(x^{\star}), h \rangle \leqslant 0 \right\}$.