Stability of closed characteristics on compact convex hypersurfaces in R^{2n}

Let $\Sigma\subset \R^{2n}$ with $n\geq2$ be any $C^2$ compact convex hypersurface and only has finitely geometrically distinct closed characteristics. Based on Y.Long and C.Zhu 's index jump methods \cite{LoZ1}, we prove that there are at least two geometrically distinct elliptic closed characteristics, and moreover, there exist at least $\varrho_{n} (\Sigma)$ ($\varrho_{n}(\Sigma)\geq[\frac{n}{2}]+1$) geometrically distinct closed characteristics such that for any two elements among them, the ratio of their mean indices is irrational number.