Based on the work of Okounkov ([15], [16]), Lazarsfeld and Mustaţȃ ([13]) and Kaveh and Khovanskii ([10]) have independently associated a convex body, called the Okounkov body, to a big divisor on a smooth projective variety with respect to a complete flag. In this paper we consider the following question: what can be said about the set of convex bodies that appear as Okounkov bodies? We show first that the set of convex bodies appearing as Okounkov bodies of big line bundles on smooth projective varieties with respect to admissible flags is countable. We then give a complete characterisation of the set of convex bodies that arise as Okounkov bodies of R-divisors on smooth projective surfaces. Such Okounkov bodies are always polygons, satisfying certain combinatorial criteria. Finally, we construct two examples of non-polyhedral Okounkov bodies. In the first one, the variety we deal with is Fano and the line bundle is ample. In the second one, we find a Mori dream space variety such that under small perturbations of the flag the Okounkov body remains non-polyhedral.

We define and study the vanishing sequence along a real valuation of sections of a line bundle on a projective variety. Building on previous work of the first author with Huayi Chen, we prove an equidistribution result for vanishing sequences of large powers of a big line bundle, and study the limit measure. In particular, the latter is described in terms of restricted volumes for divisorial valuations. We also show on an example that the associated concave function on the Okounkov body can be discontinuous at boundary points.Comment: 17 pages, supercedes the preprint arXiv:1210.3523. v2: 21 pages, expanded Section 2. Final version, to appear in Math. An

In this paper we relate the SHGH Conjecture to the rationality of onepoint Seshadri constants on blow ups of the projective plane, and explain how rationality of Seshadri constants can be tested with the help of functions on Newton-Okounkov bodies.