We will investigate the subadditivity theorem of Okounkov bodies for algebraic fiber spaces. As an application, we obtain the subadditivity of the numerical Kodaira dimension and the restricted volume for algebraic fiber spaces. As a byproduct, we obtain a criterion of birational isotriviality in terms of Okounkov bodies when the general fiber is of general type. As a special case, we prove some variants of the Iitaka conjecture. We expect that our results will provide a new approach toward the Iitaka conjecture.
This is an ongoing research, in collaboration with Dr. Jinhyung Park.