Polygons of Newton-Okounkov type on irreducible holomorphic symplectic manifolds

Let \(X\) be a projective irreducible holomorphic symplectic manifold. We associate with any big \(\mathbf{R}\)-divisor \(D\) on \(X\) a convex polygon \(\Delta_E^{\mathrm{num}}(D)\) of dimension \(2\), whose Euclidean volume is \(\mathrm{vol}_{\mathbf{R}^2}(\Delta_E^{\mathrm{num}}(D))=q_X(P(D))/2\), where \(E\) is any prime divisor on \(X\), \(q_X\) is the Beauville-Bogomolov-Fujiki form and \(P(D)\) is the positive part of the divisorial Zariski decomposition of \(D\). We systematically study these polygons and observe that they behave like the Newton-Okounkov bodies of big divisors on smooth complex projective surfaces, with respect to a general admissible flag.