Closed characteristics on compact convex hypersurfaces in $\R^8$

Abstract

In this paper, we prove there exist at least four geometrically distinct closed characteristics on every compact convex hypersurface $\Sg$ in $\R^8$. This gives a confirmed answer in the case $n=4$ to a long standing conjecture in Hamiltonian analysis since the time of A. M. Liapounov in 1892 (cf. P. 235 of \cite{Eke3}).