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 ${\mathrm{\Delta }}_E^{\mathrm{n}\mathrm{u}\mathrm{m}}(D)$ of dimension 2, whose Euclidean volume is ${\mathrm{v}\mathrm{o}\mathrm{l}}_{{\mathbf{R}}^2}({\mathrm{\Delta }}_E^{\mathrm{n}\mathrm{u}\mathrm{m}}(D))=q_X(P(D))\mathrm{/}2$, 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.