The least Steklov eigenvalue d1 for the biharmonic operator
in bounded domains gives a bound for the positivity preserving property for the hinged
plate problem, appears as a norm of a suitable trace operator, and gives the optimal
constant to estimate the L2-norm of harmonic functions. These
applications suggest to address the problem of minimizing d1
in suitable classes of domains. We survey the existing results and conjectures about this
topic; in particular, the existence of a convex domain of fixed measure minimizing
d1 is known, although the optimal shape is still unknown. We
perform several numerical experiments which strongly suggest that the optimal planar shape
is the regular pentagon. We prove the existence of a domain minimizing
d1 also among convex domains having fixed perimeter and
present some numerical results supporting the conjecture that, among planar domains, the
disk is the minimizer.