Über scharfe Schattengrenzen und Schnitte konvexer Polytope. (Sharp shadow boundaries and sections of convex polytopes)
In the paper of \textit{P. Kleinschmidt} and \textit{U. Pachner} in Mathematika 27, 58-63 (1980; Zbl 0417.52009) a sharp upper bound for the number of sharp shadow boundaries of convex d-polytopes P under central illumination from q are given; a lower bound is not yet known. (A sharp shadow boundary occurs if and only if the lines of support through q intersect P in exactly one point). Here the author finds sharp upper and lower bounds in the case of parallel illumination. As a tool he uses Grünbaum's projective (d-1)-arrangements. Theorem 1: If \(a_ p(m,d)\) denotes the number of sharp shadow boundaries of a convex polytope with m facet normals under parallel illumination, then \(2^{d-2}(m-d+2) \leq a_ p(m,d) \leq \sum^{d-1}_{i=0}\left( \begin{matrix} m-1\\ i\end{matrix} \right).\) Corollary: A convex polytope is a parallelotope if and only if it has \(2^{d-1}\) sharp shadow boundaries under parallel illumination. Theorem 2: A convex polytope is a simplex if and only if the numbers of sharp shadow boundaries for central and for parallel illumination are equal. Using polarity, the author then formulates the corresponding results about the number of partitions of the vertex set of a convex polytope by hyperplanes. Finally, he points out an interesting connection with the projection body [see \textit{T. Bonnesen} and \textit{W. Fenchel}: ''Theorie der konvexen Körper.'' (1934; Zbl 0008.07708)].
sharp shadow boundaries of convex d-polytopes
scientific article; zbMATH DE number 3933913