The main focus of this dissertation is on exploring methods to characterize the diagonals of projections in matrix algebras over von Neumann algebras. This may be viewed as a non-commutative version of the generalized Pythagorean theorem and its converse (Carpenter's Theorem) studied by R. Kadison. A combinatorial lemma, which characterizes the permutation polytope of a vector in $\mathbb{R}^n$ in terms of majorization, plays an important role in a proof of the Schur-Horn theorem. The Pythagorean theorem and its converse follow from this as a special case. In the quest for finding a non-commutative version of the lemma alluded to above, the notion of C*-convexity looks promising as the correct generalization for convexity. We make generalizations and improvements of some results known about C*-convex sets. We prove the Douglas lemma for von Neumann algebras and use it to prove some new results on the one-sided ideals of von Neumann algebras. As a useful technical tool, a non-commutative version of the Gram-Schmidt process is proved for finite von Neumann algebras. A complete characterization of the diagonals of projections in full matrix algebras over an abelian C*-algebra is provided in chapter 5. In chapter 6, we study the problem in the case of M_2(M_n(C)), the full algebra of 2 x 2 matrices over M_n(C). The example gives us hints regarding the possibility of extracting an underlying notion of convexity for C*-polytopes, which are not necessarily convex.