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In this research, the author uses Morse theory, the index iteration theory for symplectic paths developed by Y. Long and his coworkers and a new commutative property for closed characteristics to solve the n=4 case for a long standing conjecture in Hamiltonian analysis. This book is reprinted from Advances in Mathematics, Vol 297, Wei Wang, Closed characteristics on compact convex hypersurfaces in R8, 93-148, 2016, with permission from Elsevier.