# Mixed volumes of Newton–Okounkov bodies

Let $$X$$ be a smooth irreducible projective complex variety of dimension $$n$$. Let $$X=Y_0\supseteq\cdots\supseteq Y_n$$ be an admissible flag. Consider $$n$$ line bundles $$L_1,\ldots,L_n$$ on $$X$$. Let $$\Delta(L_1),\ldots,\Delta(L_n)$$ be the corresponding Newton–Okounkov bodies. By a theorem of Jow, the numerical class of $$L_i$$ is determined by the convex body $$\Delta(L_i)$$. In particular, the movable intersection product $$\langle L_1,\ldots,L_n\rangle$$ is determined by the collection $$\Delta(L_1),\ldots,\Delta(L_n)$$ with respect to all admissible flags. I'm wondering if there is an explicit formula in general?

• I thought that in Jow's theorem you only have that if $\Delta(L)=\Delta(M)$ for all admissible flags, then $L\equiv M$, but that only having this equality for just one fixed flag (like in your question) may not be enough Nov 12, 2021 at 14:50
• @YangMills Of course. By a formula, I have in mind a formula that involves all admissible flags. In fact, I have a more precise conjecture: $\langle L_1,\ldots,L_n \rangle=n!\sup \mathrm{vol} (\Delta(L_1),\ldots,\Delta(L_n))$, where the sup is taken over all admissible flags. Nov 12, 2021 at 16:27