Lower Bounds for Simplicial Covers and Triangulations of Cubes

Authors:
Adam Bliss;Francis Edward Su
Affiliations:
Department of Mathematics, Harvey Mudd College, Claremont, CA 91711, USA;Department of Mathematics, Harvey Mudd College, Claremont, CA 91711, USA
Venue:
Discrete & Computational Geometry
Year:
2005

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Abstract

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.