Hadwiger-Wills-type higher-dimensional generalizations of Pick's theorem
Pick's classical result gives a simple closed formula for the area of a (not necessarily convex) lattice polygon in terms of the interior lattice points and the lattice points on the boundary. This has been extended to non-proper lattice polygons by \textit{H. Hadwiger} and \textit{J. M. Wills} [J. Reine Angew. Math. 280, 61-69 (1976; Zbl 0321.52004)]. A polyhedron is a triangulated point set in~\(\mathbb R^d\). It is called non-proper if the underlying simplicial complex is not pure, that is, it has facets of varying dimension. Extensions of Pick's theorem to proper polyhedra of arbitrary dimension have been obtained by \textit{I. Macdonald} [Proc. Camb. Philos. Soc. 59, 719-726 (1963; Zbl 0126.18103)] and others. The author of the paper under review proves formulae for the volume of non-proper lattice polyhedra in dimensions three and four.
scientific article; zbMATH DE number 1507681
scientific article