In this paper, we prove the existence of at least n + 3 2 n+32 geometrically distinct closed characteristics on every compact non-degenerate star-shaped hypersurface Σ in R 2 n {R}^2n with odd n ≥ 3 n 3 provided that every prime closed characteristic possesses nonzero mean index, and the Maslov-type index of every closed characteristic (including iterations) is not equal to − 1 -1.
Duan et al. (Tue,) studied this question.