Brake type closed characteristics on reversible compact convex hypersurfaces in R2n

In this paper, let ΣΣ be a C3C3 compact convex in R2n satisfying the reversible condition NΣ=ΣNΣ=Σ with N=diag(−In,In). We prove that if there are exactly nn geometrically distinct closed characteristics on ΣΣ and all of them are nondegenerate, then all of them must be brake orbits up to a suitable translation of time. Moreover, for n=2n=2 or 3, we prove that if there are exactly nn geometrically distinct closed characteristics on ΣΣ, then all of them must be brake orbits up to a suitable translation of time.