Closed characteristics on compact convex hypersurfaces in R8

In this paper, we prove there exist at least four geometrically distinct closed characteristics on every compact convex hypersurface Σ in R8R8. This gives a confirmed answer in the case n=4n=4 to a long standing conjecture in Hamiltonian analysis since the time of A. M. Liapounov in 1892 (cf. P. 235 of [4]).