  We deal with the following eigenvalue optimization problem: Given a bounded
domain $D\subset \R^2$, how to place an obstacle $B$ of fixed shape within $D$
so as to maximize or minimize the fundamental eigenvalue $\lambda_1$ of the
Dirichlet Laplacian on $D\setminus B$. This means that we want to extremize the
function $\rho\mapsto \lambda_1(D\setminus \rho (B))$, where $\rho$ runs over
the set of rigid motions such that $\rho (B)\subset D$. We answer this problem
in the case where both $D$ and $B$ are invariant under the action of a dihedral
group $\mathbb{D}_n$, $n\ge2$, and where the distance from the origin to the
boundary is monotonous as a function of the argument between two axes of
symmetry. The extremal configurations correspond to the cases where the axes of
symmetry of $B$ coincide with those of $D$.
