This thesis mainly focuses on geometric measure theory with applications to some shape optimization problems being considered over rough sets. We extend previous theory of traces for rough vector fields over rough domains and proved the compactness of uniform domains without uniformly bounded perimeter assumption. As application of these results, together with some other tools from geometric analysis, we can give partial results on the existence and uniqueness of minimizers of the nematic liquid droplets problem and the thermal insulation problem. Two other geometric minimization problems with averaged property, which include the generalized Cheeger set problem, are also studied.