  In this paper we study the existence and non-existence of minimizers for a
type of (critical) Poincar\'{e}-Sobolev inequalities. We show that minimizers
do exist for smooth domains in $\mathbb{R}^d$, an also for some polyhedral
domains. On the other hand, we prove the non-existence of minimizers in the
rectangular isosceles triangle in $\mathbb{R}^2$.
