  We establish the relationship between folded symplectic forms and convex
hypersurface theory in contact topology. As an application, we use convex
hypersurface theory to reprove and strengthen the existence result for folded
symplectic forms due to Cannas da Silva, and we generalize to all even
dimensions Baykur's $4$-dimensional existence result of folded Weinstein
structures and folded Lefschetz fibrations.
