Lower Bounds for Simplicial Covers and Triangulations of Cubes Adam Bliss ' 03

We show that the size of a minimal simplicial cover of a polytope P is a lower bound for the size of a minimal triangulation of P , including ones with extra vertices. We then use this fact to study minimal triangulations of cubes, and we improve lower bounds for covers and triangulations in dimensions 4 through at least 12 (and possibly more dimensions as well). Important ingredients are an analysis of the number of exterior faces that a simplex in the cube can have of a specified dimension and volume, and a characterization of corner simplices in terms of their exterior faces. Let P be a convex polytope. A (simplicial) cover of P is a collection of simplices such that (i) the vertices of the simplices are vertices of P and (ii) the union of the simplices is P . For example, some covers are triangulations in which simplices meet face-to-face and have disjoint interiors; although in general, cover elements may overlap. Define the covering number C(P ) to be the minimal number of simplices needed for a cover of a polytope P . Although the covering number is of interest in its own right (see [3]), we prove in Theorem 1 that the covering number of P also gives a lower bound for the size of a minimal triangulation of P , including triangulations with extra vertices. We then use the covering number to study the classical problem of determining the size of the minimal triangulation when P is a d-dimensional cube (this is sometimes called the simplexity of the cube). Let I = [0, 1], and let I denote the d-cube. We define the notion of an exterior face of a cube simplex, develop a counting function for exterior faces of prescribed dimension and volume, and establish a recursive bound in Theorem 2 that yields an absolute bound for the cases of interest in Theorem 3. This bound can be improved still further by analyzing separately the corner simplices of the cube; Theorem 4 characterizes corner simplices in terms of the number of exterior faces they have. We use these results to establish new lower bounds for C(I), and via Theorem 1, these yield the best known bounds for general triangulations in dimensions up through at least 12 (and possibly more dimensions— see the concluding section). For d = 4 through d = 12 these numbers are: 16, 60, 252, 1143, 5104, 22616, 98183, 520865, 2.9276 × 10 In particular, our results show that Mara’s minimal triangulation of the 4-cube [7], using 16 simplices, is also a minimal simplicial cover; furthermore, using extra vertices will not produce any smaller triangulations of the 4-cube. See Theorem 5. A comparison of our bounds with bounds for other kinds of decompositions may be found in Table 1. ∗Research partially supported by a Beckman Research Grant at Harvey Mudd College. ∗∗Research partially supported by NSF Grant DMS-0301129. 1 2 ADAM BLISS AND FRANCIS SU This paper is organized as follows. Section 1 discusses the relationship between minimal simplicial covers and minimal triangulations of arbitrary polytopes P . Section 2 gives background on triangulations of cubes. We develop some terminology in Section 3, and in Section 4, we develop constraints for a linear program for our problem. This involves a counting function F which counts the number of exterior faces that a simplex can have; in order to estimate F , we develop in Sections 5 and 6 some theory regarding the way exterior faces of simplices relate to simplices in the cube. Then, in Section 7 we show how this theory produces a recurrence for the counting function that can be used to get bounds on F , and Section 8 refines the earlier linear program. The final section discusses our results and some open questions. 1. Minimal Covers bound Triangulations with extra vertices When speaking of minimal triangulations of a polytope P , we must be careful to distinguish what kind of decomposition we mean, since there are many such notions in the literature. In this paper, a triangulation of a d-polytope P will always mean a decomposition of P into d-simplices with disjoint interiors, such that each pair of simplices intersect in a face common to both or not at all. In the literature, some authors restrict attention to triangulations whose vertices are required to be vertices of the polytope. We refer to these as vertex triangulations. Others do not always require triangulations to meet face-to-face; such decompositions are sometimes called dissections. They are covers by simplices that do not have overlapping interiors. A vertex dissection is a dissection in which vertices of simplices come from those of the polytope. We let D(P ), T (P ),D(P ), T (P ) denote, respectively, the size of the smallest possible dissection, triangulation, vertex dissection, and vertex triangulation of P . (We note that our notation differs from that of Smith [12], who uses T to include dissections, and of Hughes-Anderson [6], who use T for vertex triangulations.) The evident inclusions immediately imply that D(P ) ≤ T (P ) ≤ T (P ) and D(P ) ≤ D(P ) ≤ T (P ). Also, C(P ) ≤ D(P ) since any vertex dissection is a simplicial cover— this does not hold for general dissections, since in a cover of P , vertices of simplices must be vertices of P . What may therefore be somewhat surprising is the following important inequality relating covers to general triangulations: Theorem 1. For any convex polytope P , the covering number C(P ) satisfies: C(P ) ≤ T (P ). Some covers arise as images of simplicial Brouwer self-maps of a triangulated polytope and are related to the polytopal Sperner lemma of DeLoera-Peterson-Su [3]. Such considerations form the basis for a proof of Theorem 1. Proof. Let K be a triangulation of P . For any such triangulation construct a piecewise linear (PL) map fK : P → P in the following way. If v is a vertex of K, define fK(v) to be LOWER BOUNDS FOR SIMPLICIAL COVERS AND TRIANGULATIONS OF CUBES 3 any vertex of P on the smallest-dimensional P -face that contains v. Then extend this map linearly across each simplex of K. See Figure 1. Thus fK is a PL-map from P to P that takes simplices of K to simplices formed by vertices of P , and these images must be a cover of P because this map is a Brouwer map of degree 1 [3, Prop. 3]. Thus there are at least C(P ) such simplices in the triangulation K.