Symmetric closed characteristics on symmetric compact convex hypersurfaces in $\mathbf{R}^8$: Special cases

Abstract

Let $\Sigma$ be a $C^3$ compact symmetric convex hypersurface in $\mathbf{R}^{8}$. For some special cases, we prove that when $\Sigma$ carries exactly four geometrically distinct closed characteristics, then all of them must be symmetric.