Multipolytopes and convex chains

Abstract

For a simple complete multipolytope \(\mathcal{P}\) in ℝn, Hattori and Masuda defined a locally constant function \(DH_\mathcal{P} \) on ℝn 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 \(DH_\mathcal{P} \) and construct a convex chain \(\overline {DH_\mathcal{P} } \) on ℝn. 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 ⨑ub;simple semicomplete multipolytopes⫂ub; →; ⨑ub;convex chains⫂ub; is surjective but not injective. We will study its “kernel.”