Let $L = \dot{q}^T M(q) \dot{q} + V(q)$, i.e., the lagrangian has a quadratic form and hence is convex w.r.t to the velocities, considering that $V(q)$ plays the role of a constant. And now let the discrete lagrangian, i.e, the lagrangian density be:
$$ L_d(q_{k}, q_{k+1}, h) = \int_{0}^{h} L(q(t), \dot{q}(t)) \,dt $$
Where $q_{k} = q(0)$ and $q_{k+1} = q(h)$. Can we infer any convex property from the discrete lagrangian with respect to any variable input $q_{k}$ or $q_{k+1}$ and if so how can I prove it?
I know that if $L$ is regular then $L_d$ has to be also regular and regularity implies that the determinant of the matrix $\frac{\partial L_d^2}{\partial q_k \partial q_{k+1}}$ is non-singular but that doesn't imply that the Hessian is positive semidefinite and that $L_d$ is convex or even concave right?.