  In this paper, we give three different new proofs of the validity of the
geometry conjecture about cycles of projections onto nonempty closed, convex
subsets of a Hilbert space. The first uses a simple minimax theorem, which
depends on the finite dimensional Hahn-Banach theorem. The second uses Fan's
inequality, which has found many applications in optimization and mathematical
economics. The third uses three results on maximally monotone operators on a
Hilbert space.
