Let $D_1, D_2$ two open, bounded and convex domain in $R^n$. Suppose that $D_2 \supset \overline{D_1}$, and the boundaries of these sets are of class $C^1$.
Fix $x \in \partial D_1$ and suppose that the hiperplane $\{x = (x_1,...,x_n) \in R^n ; x_1 = 0\}$ is tangent at $x$ and $D_1 \subset \{x = (x_1,...,x_n) \in R^n ; x_1 < 0\} $
Clearly we can find $y_x \in A:= \partial D_2 \cap \{ z ; (z-x).e_1 >0\}$ such that
$$ max_{z \in A} (z-x).e_1 = (y_x - x). e_1$$
Intuitively $y_x = (\alpha,0,...,0)$ for some $\alpha \in R$ and the hyperplane tanget at $y_x$ is paralalel to the hiperplane $\{x = (x_1,...,x_n) \in R^n ; x_1 = 0\}$. This is true?
I made these questions because appears that these intuitive affirmations is used in the proof of the lemma 2.2 of this paper
Existence of classical solutions to a free boundary problem for the p-Laplace operator, (I) by A. Henrot and H. Shahgholian
I dont know how to prove .. I believe that is the thing of the type: intuitive to see and hard to prove.
Someone could help me, please?
thanks in advance