  In this paper, let $\Sigma\subset\R^{6}$ be a compact convex hypersurface. We
prove that if $\Sigma$ carries only finitely many geometrically distinct closed
characteristics, then at least two of them must possess irrational mean
indices. Moreover, if $\Sg$ carries exactly three geometrically distinct closed
characteristics, then at least two of them must be elliptic.
