  In this manuscript we study the following optimization problem: given a
bounded and regular domain $\Omega\subset \mathbb{R}^N$ we look for an optimal
shape for the "$\mathrm{W}-$vanishing window" on the boundary with prescribed
measure over all admissible profiles in the framework of the Orlicz-Sobolev
spaces associated to constant for the "Sobolev trace embedding". In this
direction, we establish existence of minimizer profiles and optimal sets, as
well as we obtain further properties for such extremals. Finally, we also place
special emphasis on analyzing the corresponding optimization problem involving
an "$\mathrm{A}-$vanishing hole" (inside the domain) with volume constraint.
