Multiple brake orbits on compact convex symmetric reversible hypersurfaces in R2nR2n

In this paper, we prove that there exist at least [n+12]+1 geometrically distinct brake orbits on every C2C2 compact convex symmetric hypersurface Σ   in R2nR2n for n⩾2n⩾2 satisfying the reversible condition NΣ=ΣNΣ=Σ with N=diag(−In,In)N=diag(−In,In). As a consequence, we show that there exist at least [n+12]+1 geometrically distinct brake orbits in every bounded convex symmetric domain in RnRn with n⩾2n⩾2 which gives a positive answer to the Seifert conjecture of 1948 in the symmetric case for n=3n=3. As an application, for n=4 and 5n=4 and 5, we prove that if there are exactly n geometrically distinct closed characteristics on Σ, then all of them are symmetric brake orbits after suitable time translation.