This dissertation has two topics. The rst one is about rotating a supporting hyperplane on the convex hull of a nite point set to arrive at one of its facets. We present three procedures for these rotations in multiple dimensions. The rst two procedures rotate a supporting hyperplane for the polytope starting at a lower dimensional face until the support set is a facet. These two procedures keep current points in the support set and accumulate new points after the rotations. The rst procedure uses only algebraic operations. The second procedure uses LP. In the third procedure we rotate a hyperplane on a facet of the polytope to a dierent adjacent facet. Similarly to the rst procedure, this procedure uses only algebraic operations. Some applications to these procedures include data envelopment analysis (DEA) and integer programming. The second topic is in the eld of containment problems for polyhedral sets. We present three procedures to nd a circumscribing simplex that contains a point set in any dimension. The rst two procedures are based on the supporting hyperplane rotation ideas from the rst topic. The third circumscribing simplex procedure uses polar cones and other geometrical properties to nd facets of a circumscribing simplex. One application of the second topic discussed in this dissertation is in hyperspectral unmixing.