  We consider the problem of minimizing convex combinations of the first two
eigenvalues of the Dirichlet-Laplacian among open sets of $R^N$ of fixed
measure. We show that, by purely elementary arguments, based on the minimality
condition, it is possible to obtain informations on the geometry of the
minimizers of convex combinations: we study, in particular, when these
minimizers are no longer convex, and the optimality of balls. As an application
of our results we study the boundary of the attainable set for the Dirichlet
spectrum. Our techniques involve symmetry results \`a la Serrin, explicit
constants in quantitative inequalities, as well as a purely geometrical
problem: the minimization of the Fraenkel 2-asymmetry among convex sets of
fixed measure.
