  In this paper, we prove that there exist at least $n$ geometrically distinct
brake orbits on every $C^2$ compact convex symmetric hypersurface $\Sg$ in
$\R^{2n}$ satisfying the reversible condition $N\Sg=\Sg$ 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 $n$.
