Geodesic
Multipolytopes and Convex Chains
Trudy Matematicheskogo Instituta imeni V.A. Steklova, Geometric topology, discrete geometry, and set theory, Tome 252 (2006), pp. 224-236 Cet article a éte moissonné depuis la source Math-Net.Ru

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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.” 
@article{TM_2006_252_a18,
     author = {Y. Nishimura},
     title = {Multipolytopes and {Convex} {Chains}},
     journal = {Trudy Matematicheskogo Instituta imeni V.A. Steklova},
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     language = {en},
     url = {http://geodesic.mathdoc.fr/item/TM_2006_252_a18/}
}
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Y. Nishimura. Multipolytopes and Convex Chains. Trudy Matematicheskogo Instituta imeni V.A. Steklova, Geometric topology, discrete geometry, and set theory, Tome 252 (2006), pp. 224-236. http://geodesic.mathdoc.fr/item/TM_2006_252_a18/

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