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.