Lower bounds on cube simplexity

Lower Bounds for Simplicial Covers and Triangulations of Cubes

“…It should be possible to improve our bounds for covers by using more information about the kinds of simplices that can occur in the cube and how they fit together, similar to what Hughes [5] and Hughes and Anderson [6] did to study (D v , T v )-minimal triangulations. These programs rely on enumerating specific features of configuration classes of simplices in the cube, or on enumerating the classes themselves.…”

“…Hughes [5], Hughes and Anderson * [6], Smith [12] Our bounds Cottle * * [2] and Sallee * * [10] Dimension …”

“…Instead, we develop a linear program whose optimal solution is a bound for the size of the minimal cover. The linear programming approach was initiated by Sallee [9], Hughes [5], and Hughes and Anderson [6] to study triangulations of the cube. However, the results of Hughes and Hughes and Anderson are not fully comparable with ours, because their methods do not apply to covers.…”

A Lower Bound for the Simplexity of then-Cube via Hyperbolic Volumes

“…Hughes [9] was able to obtain a lower bound on T v (n) for small n by solving enormous linear or integer programs, in which the variables corresponded to exterior j-face tuples, with computer aid. These methods are remarkably powerful for small n, actually obtaining the apparently correct answers when n ≤ 7, and (as Table 1 shows) yielding bounds at least as good as the present paper's for n ≤ 11, i.e., all n for which Hughes's computer was able to TABLE 1.…”

“…Most of them come from exhaustive searches, computer-aided constructions, and/or bounds obtained from enormous linear or integer programs [9,12].…”