In this paper, we prove that there exist at least n+1 2 + 1 geometrically distinct brake orbits on every C 2 compact convex symmetric hypersurface Σ in R 2n for n ≥ 2 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 + 1 geometrically distinct brake orbits in every bounded convex symmetric domain in R n with n ≥ 2 which gives a positive answer to the Seifert conjecture of 1948 in the symmetric case for n = 3. As an application, for n = 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.