I am reading the book "Set-Valued analysis" by Jean-Pierre Aubin and Hèlène Frankowska. On page 152, table 4.4, statement 5b), we read:
If $K_1$ and $K_2$ are closed derivable subsets contained in $X, x \in K_1 \cap K_2$ satisfies $C_{K_1}(x)-C_{K_2}(x)=X$, then $T_{K_1 \cap K_2}(x)=T_{K_1}(x) \cap T_{K_2}(x)$.
The proof is not given, so I am trying to do it on my own. I am struggling, however. $T_{({\cdot})}$ and $C_{({\cdot})}$ are the Bouligand contingent cone and the Clarke tangent cone respectively.
Let me define all these terms. I am only showing the definitions in terms of sequences, because I think that they are the ones that should be used in the proof.
- $v$ belongs to $T_{K}(x)$ if and only if there exists a sequence of $h_n>0$ converging to $0+$ and a sequence of $v_n \in X$ converging to $v$ such that $$x+h_n v_n \in K \quad \forall n \geq 0.$$
- $v$ belongs to $C_{K}(x)$ if and only if for every sequence $h_n$ converging to $0+$ and every sequence $x_n$ converging to $x$ with $x_n \in K$ there exists a sequence of $v_n \in X$ converging to $v$ such that $$x_n+h_n v_n \in K \quad \forall n \geq 0.$$
- A subset $K$ is called derivable at $x$ iff $T_{K}(x) = T^b_K(x)$, where $T^b_K(x)$ is given by:
$$v \in T^b_K(x) \Leftrightarrow \forall h_n \to 0^+, \; \exists v_n \to v \text{ such that } \forall n, \quad x + v_n h_n \in K$$
The inclusion $T_{K_1 \cap K_2}(x) \subset T_{K_1}(x) \cap T_{K_2}(x)$ is straightforward and always holds, so we only have to prove $T_{K_1}(x) \cap T_{K_2}(x) \subset T_{K_1 \cap K_2}(x)$.
Therefore, we want to prove that for any $v \in T_{K_1}(x) \cap T_{K_2}(x)$, it holds that $v \in T_{K_1 \cap K_2}(x)$. That is, we want to find a sequence $h_n \to 0^+$ and a sequence $v_{n} \to v$ such that $x + h_n v_n \in K_1 \cap K_2$ for all $n \geq 0$.
I really don't know how to proceed; the fact that we are requiring $C_{K_1}(x)-C_{K_2}(x)=X$ makes me think that I should write $v$ as $v = v_1 -v_2$, where $v_1 \in C_{K_1}(x)$ and $v_2 \in C_{K_2}(x)$, but it does not lead me anywhere.
I don't need a detailed proof. Maybe just a hint is enough to "unstuck" me.
This question is somehow related, but I don't think the answer helps me, as the sets in our case are not necesarilly convex.
