Multipolytopes and Convex Chains

For a simple complete multipolytope $\mathcal P$ in $\mathbb R^n$, Hattori and Masuda defined a locally constant function $\mathrm {DH}_{\mathcal P}$ on $\mathbb R^n$ minus the union of hyperplanes associated with $\mathcal P$, which agrees with the density function of an equivariant complex line bundle over a Duistermaat–Heckman measure when $\mathcal P$ arises from a moment map of a torus manifold. We improve the definition of $\mathrm {DH}_{\mathcal P}$ and construct a convex chain $\overline {\mathrm {DH}}_{\mathcal P}$ on $\mathbb R^n$. The well-definiteness of this convex chain is equivalent to the semicompleteness of the multipolytope $\mathcal P$. Generalizations of the Pukhlikov–Khovanskii formula and an Ehrhart polynomial for a simple lattice multipolytope are given as corollaries. The constructed correspondence $\{$simple semicomplete multipolytopes$\}\to \{$convex chains$\}$ is surjective but not injective. We will study its “kernel.”