I want to prove a theorem (the link is after the whole text here) and in order to do that I need to prove three preliminary statements. I tried to prove them all but I'm already stuck in the first, and this one seems to be necessary to prove the others. I hope someone here can help me on this, thanks.
We have two compact convex subsets $A_1, A_2$ of $\mathbb{R}^n$, also assume that the origin is in the interior of both subsets. Denote $E_1$ for the linear hull of $A_1$, same thing for $E_2$. Furthermore, for any $x_0\in A_1$, denote $C_{x_0} = \{y\in E_2: \ \forall x\in A_1, \ \langle y, x_0-x\rangle \geq 0\}$. Finally, consider $\lambda_1,\lambda_2 >0$ two arbitrary and fixed real numbers and consider the set $A = \lambda_1A_1+\lambda_2A_2$. The three steps to prove the theorem are:
i) Show that for any $a\in A$ there are $x\in A_1$ and $y\in C_x\cap A_2$ such that $a = \lambda _1 x+ \lambda_2 y$.
ii) This decomposition is unique.
iii) The map $a\mapsto (x,y)$ described above is Lipschitz and smooth almost everywhere.
My attempt: Ok, for the first one I know than $a$ can be written as $a=\lambda_1 x + \lambda_2 y$ for some $x\in A_1, y\in A_2$, but it's possible that $y\notin C_x\cap A_2$. My idea was to consider elements like $ty$, $t\in\mathbb{R}$, which are elements in $C_x$, and knowing that $0\in A$, we could try to take smalls values for $t$ to get elements in $A_2$, one of them will fits the properties desired. But the way to show this is nothing clear.
About the third item, to show that the map is smooth almost everywhere, I can use a result which says that if a function is Lipschitz in an open subset of $\mathbb{R}^n$, then this function is smooth almost everywhere. Therefore, I want to show there is some convenient open subset containing $A$. I just don't know how to show this.
By the way, this is for proving this theorem, the part which says One can show that f is a homogeneous polynomial of degree n, therefore it can be written as...
Thank you!