Extremal properties for dissections of convex 3-polytopes (Q2706196)

A dissection of a convex \(d\)-polytope is a partition into \(d\)-simplices with disjoint interiors; in a triangulation, these simplices meet face-to-face. (Of course, a triangulation is a dissection.) More generally, one can consider dissections or triangulations, whose simplices fall into a given finite set \(\mathcal A\). In the cases under discussion, \(\mathcal A\) is the vertex-set of the polytope.NEWLINENEWLINENEWLINEThe topic of the paper is mismatches between the maximal or minimal numbers of simplices in dissections as opposed to triangulations. When \(d = 3\), the authors prove several somewhat surprising results. For instance, with the maximal problem for simplicial 3-polytopes, the mismatch can be linear in the number of vertices. Further, there is a 3-polytope with 8 vertices, for which the mismatches are positive in both maximal and minimal cases.NEWLINENEWLINENEWLINEThe paper also contains results about triangulations of combinatorial cubes and Archimedean 3-polytopes.