Neumann eigenvalues having no mononicity property with respect to domain inclusion, it makes sense to study the following shape optimization problems: \begin{eqnarray} \min \{ \mu_k(\Omega)\;:\; \Omega: \text{ convex}, \;\Omega \subset D\} \quad \text{(for a given box $D$)},\\ \max\{\mu_k(\Omega) \;:\; \Omega: \text{convex}, \; \omega \subset \Omega \} \quad \text{(for a given obstacle $\omega$)}. \end{eqnarray} In this paper, we study the existence of a solution for these two problems in two dimensions, and we give some qualitative properties. We also introduce the notion of self-domains that are domains solutions of these extremal problems for themselves, and give examples of the disk and the square.