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**On the combinatorial structure of \(0/1\)-matrices representing nonobtuse simplices.**
*(English)*
Zbl 1524.05298

A \((0,1)\)-simplex is the convex hull of \(n+1\) affinely independent vertices of the \(n\) dimensional unit cube. It is nonobtuse if none of its dihedral angles is obtuse, and it is acute if its dihedral angles are all acute. An orthogonal simplex is a nonobtuse simplex whose dihedral angles include exactly \(n\) acute angles and \(\frac12n(n-1)\) right angles. This paper studies the \((0,1)\)-matrices that represent acute, nonobtuse, and (to a lesser extent) orthogonal simplices.

The simplices are studied up to the action of the hyperoctahedral group of symmetries of the cube. For this reason, it may be assumed that the origin is one of the vertices. A matrix representation of the simplex can then be obtained, with each column listing the coordinates of one of the other \(n\) vertices.

It is shown that if \(P\) is a matrix representation of an acute \((0,1)\)-simplex then \(P\) has a fully indecomposable doubly stochastic pattern. The corresponding result for nonobtuse simplices is that the support of \(P\) must contain a doubly stochastic pattern (and need not be fully indecomposable). The smallest fully indecomposable nonobtuse \((0,1)\)-simplices that are not acute are found (computationally) to have dimension 9.

The main result of the last section is this: Let \(S\) be a nonobtuse \((0,1)\)-simplex whose fully indecomposable components are all acute. Then \(S\) has at most one face-to-face neighbor at each of its interior facets. This extends the so-called “one neighbor theorem” for acute simplices, proved in [J. Brandts et al., Comput. Geom. 46, No. 3, 286–297 (2013; Zbl 1261.65020)].

The simplices are studied up to the action of the hyperoctahedral group of symmetries of the cube. For this reason, it may be assumed that the origin is one of the vertices. A matrix representation of the simplex can then be obtained, with each column listing the coordinates of one of the other \(n\) vertices.

It is shown that if \(P\) is a matrix representation of an acute \((0,1)\)-simplex then \(P\) has a fully indecomposable doubly stochastic pattern. The corresponding result for nonobtuse simplices is that the support of \(P\) must contain a doubly stochastic pattern (and need not be fully indecomposable). The smallest fully indecomposable nonobtuse \((0,1)\)-simplices that are not acute are found (computationally) to have dimension 9.

The main result of the last section is this: Let \(S\) be a nonobtuse \((0,1)\)-simplex whose fully indecomposable components are all acute. Then \(S\) has at most one face-to-face neighbor at each of its interior facets. This extends the so-called “one neighbor theorem” for acute simplices, proved in [J. Brandts et al., Comput. Geom. 46, No. 3, 286–297 (2013; Zbl 1261.65020)].

Reviewer: Ian M. Wanless (Clayton)

### MSC:

05E45 | Combinatorial aspects of simplicial complexes |

05B20 | Combinatorial aspects of matrices (incidence, Hadamard, etc.) |

15B36 | Matrices of integers |

52B05 | Combinatorial properties of polytopes and polyhedra (number of faces, shortest paths, etc.) |