  In this paper, we consider the minimal period estimates for brake orbits of
nonlinear symmetric Hamiltonian systems. We prove that if the Hamiltonian
function $H\in C^2(\Bbb R^{2n}, \Bbb R)$ is super-quadratic and convex, for
every number $\tau>0$, there exists at least one $\tau$-periodic brake orbit
$(\tau,x)$ with minimal period $\tau$ or $\tau/2$ provided $H(Nx)=H(x)$.
