In this paper, we prove that there exist at least n geometrically distinct brake orbits on every C 2 compact convex symmetric hypersurface Σ in R 2n satisfying the reversible condition N Σ = Σ with N = diag(−I n , I n ). As a consequence, we show that if the Hamiltonian function is convex and even, then Seifert conjecture of 1948 on the multiplicity of brake orbits holds for any positive integer n.