Closed characteristics on compact convex hypersurfaces in ²ⁿ

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For any given compact C² hypersurface Σin R²ⁿ bounding a strictly convex set with nonempty interior, in this paper an invariant ₙ(Σ) is defined and satisfies ₙ(Σ)≥ [n/2]+1, where [a] denotes the greatest integer which is not greater than a∈ R. The following results are proved in this paper. There always exist at least ρₙ(Σ) geometrically distinct closed characteristics on Σ. If all the geometrically distinct closed characteristics on Σare nondegenerate, then ₙ(Σ)≥ n. If the total number of geometrically distinct closed characteristics on Σis finite, there exists at least an elliptic one among them, and there exist at least ₙ(Σ)-1 of them possessing irrational mean indices. If this total number is at most 2ₙ(Σ) -2, there exist at least two elliptic ones among them.