A Characterization of Gram Matrices of Polytopes

Volumes of degenerating polyhedra — on a conjecture of J. W. Milnor

“…The domain can be characterized as follows (this result is classical and has been reproved many times, the most general version of this result has been proved by Raquel Diaz-Sanchez in [1]): The domain is thus bounded by the zero sets of polynomials in the cosines of α i j -this is true for condition (b) as well as the two others, since a 2 × 2 matrix is positive definite if and only if both its trace and its determinant are positive. Since the gradients of all such polynomials are bounded by some constant C (depending only on dimension), we have the following elementary observation:…”

“…First, we give a simple argument to show a sharp version of Milnor's Continuity Conjecture for all hyperbolic polytopes of dimension greater than 3, and also all spherical polytopes. It should be noted that since in many cases it is not known whether hyperbolic or spherical polytopes are determined by their dihedral angles and how to characterize the possible assignments of dihedral angles, 1 it makes more sense to use the polar metrics introduced in [7,9]. The argument shows that the extension is, in fact, Lipschitz.…”

Spherical tetrahedra with rational volume, and spherical Pythagorean triples

“…Then S is proportional to one of the following list: . According to Theorem 2.3 there is a single continuous family of linear combinations of cosines, depending on a real-valued parameter t, which, for every instance of t ∈ πQ, provides a rational solution to (5). The remaining linear combinations we call sporadic, in order to distinguish them from continuous families.…”

“…In Monty [9] we use a brute-force search over the set of all dihedral angles with denominators from the union of the above mentioned lists L i , i ∈ {0, 1, 2, 3}. This does not result in anà priori efficient search, however turns out to be sufficient to find all sporadic solutions to (5) and, subsequently, to (2).…”

Gram Matrix Analysis of Finite Distance Spaces in Constant Curvature

“…However, the C κ -matrix can also be given another interpretation, namely as dual to the matrix of cosines to exterior dihedral angles, assuming the points of X span a simplex in the space form under consideration. This matrix of cosines is also, in the classical setting, sometimes called a Gram matrix, see [10] and [4]. See below for a clarification of these concepts.…”