For the case $K=\mathbb{C}$, statement (2) in the above result is related to the Guillemin-Abreu classification of Kähler structures on symplectic toric varieties as explained in [Abr03]. By definition, a symplectic toric variety is a compact symplectic manifold of dimension $2n$ together with a Hamiltonian action of the compact torus $\mathbb{S}^{{\text{\rm an}}}\simeq(S^{1})^{n}$. These spaces are classified by Delzant polytopes of $M_{\mathbb{R}}$, see for instance [Gui95]. For a given Delzant polytope $\Delta\subset M_{\mathbb{R}}$, the possible $(S^{1})^{n}$-invariant Kähler forms on the symplectic toric variety corresponding to $\Delta$ are classified by smooth convex functions on $\Delta^{\circ}$ satisfying some conditions near the border of $\Delta$. Several differential geometric invariants of a Kähler toric variety can be translated and studied in terms of this convex function, also called the ‘‘symplectic potential’’.

For a smooth positive toric metric $\|\cdot\|$ on $L_{\Psi_{\Delta}}(\mathbb{C})$, the Chern form defines a Kähler structure on the complex toric variety $X_{\Sigma_{\Delta}}(\mathbb{C})$. It turns out that the corresponding symplectic potential coincides with minus the function $\psi^{\vee}_{\|\cdot\|}$. It would be most interesting to explore further this connection.