Abstract
A dissection of a convex d-polytope is a partition of the polytope into d-simplices whose vertices are among the vertices of the polytope. Triangulations are dissections that have the additional property that the set of all its simplices forms a simplicial complex. The size of a dissection is the number of d-simplices it contains. This paper compares triangulations of maximal size with dissections of maximal size. We also exhibit lower and upper bounds for the size of dissections of a 3-polytope and analyze extremal size triangulations for specific non-simplicial polytopes: prisms, antiprisms, Archimedean solids, and combinatorial d-cubes.
Highlights
Let A be a point configuration in Rd with its convex hull conv (A) having dimension d
A dissection is a triangulation of A if in addition any pair of simplices intersects at a common face
We say that a dissection is mismatching when it is not a triangulation
Summary
Let A be a point configuration in Rd with its convex hull conv (A) having dimension d. In this paper we augment previous results by showing that it is possible to have simultaneously, in the same 3-polytope, that the size of a mismatching minimal (maximal) dissection is smaller (larger) than any minimal (maximal) triangulation. We show that the gap between the size of a mismatching maximal dissection and a maximal triangulation can grow linearly on the number of vertices and that this occurs already for a family of simplicial convex 3-polytopes. (1) There exists an infinite family of convex simplicial 3-polytopes with increasing number of vertices whose mismatching maximal dissections are larger than their maximal triangulations. This gap is linear in the number of vertices (Corollary 2.2). We will call the north (respectively, south) vertex of Qm the one which maximizes (respectively, minimizes) the scalar product with the vector
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