Okounkov bodies associated to pseudoeffective divisors II

We first prove some basic properties of Okounkov bodies, and give a characterization of Nakayama and positive volume subvarieties of a pseudoeffective divisor in terms of Okounkov bodies. Next, we show that each valuative and limiting Okounkov bodies of a pseudoeffective divisor which admits the birational good Zariski decomposition is a rational polytope with respect to some admissible flag. This is an extension of the result of Anderson-K\"{u}ronya-Lozovanu about the rational polyhedrality of Okounkov bodies of big divisors with finitely generated section rings.
Comment: 14 pages, deleted previous Section 5 and fixed several gaps in the previous version. to appear in Taiwanese J. Math. (special issue for the proceedings of the conference Algebraic Geometry in East Asia 2016)