Multiplicity of brake orbits on compact convex symmetric reversible hypersurfaces in R 2 n for n ⩾4

In this paper, we prove that there exist at least [( n +1)/2]+2 geometrically distinct brake orbits on every C 2 compact convex symmetric hypersurface Σ in R 2 n for n ⩾4 satisfying the reversible condition N Σ=Σ with N =diag (− I n , I n ). As a consequence, we show that there exist at least [( n +1)/2]+2 geometrically distinct brake orbits in every bounded convex symmetric domain in R n with n ⩾4. For n =4, 5, this result gives a positive answer to the Seifert conjecture of 1948 in the convex symmetric case. For n =3, a corresponding positive answer was given in Zhang and Liu [‘Multiple brake orbits on compact convex symmetric reversible hypersurfaces in R 2 n ’, arXiv:1111.0722vl [math. SG]], and n ⩽2 a corresponding positive answer was given in Long et al. [‘Multiple brake orbits in bounded convex symmetric domains’, Adv. in Math. 203 (2006) 568–635] and in Liu and Zhang [‘Iteration theory of L ‐index and multiplicity of brake orbits’, arXiv: 0908.0021vl [math. SG]].