No abstract

Given a degenerate (n+1)-simplex in a d-dimensional space M d (Euclidean, spherical or hyperbolic space, and d ≥ n), for each k, 1 ≤ k ≤ n, Radon's theorem induces a partition of the set of k-faces into two subsets. We prove that if the vertices of the simplex vary smoothly in M d for d = n, and the volumes of k-faces in one subset are constrained only to decrease while in the other subset only to increase, then any sufficiently small motion must preserve the volumes of all k-faces; and this property still holds in M d for d ≥ n + 1 if an invariant c k−1 (α k−1 ) of the degenerate simplex has the desired sign. This answers a question posed by the author, and the proof relies on an invariant c k (ω) we discovered for any k-stress ω on a cell complex in M d . We introduce a characteristic polynomial of the degenerate simplex by defining f (x) = n+1 i=0 (−1) i c i (α i )x n+1−i , and prove that the roots of f (x) are real for the Euclidean case. Some evidence suggests the same conjecture for the hyperbolic case.

Let M d be the spherical, Euclidean, or hyperbolic space of dimension d ≥ n + 1. Given any degenerate (n + 1)-simplex A in M d with non-degenerate n-faces F i , there is a natural partition of the set of n-faces into two subsets X 1 and X 2 such that, except for a special spherical case where X 2 is the empty set and X1 V n (F i ) = V n (S n ) instead. For all cases, if the vertices vary smoothly in M d with a single volume constraint that X1 V n (F i ) − X2 V n (F i ) is preserved as a constant (0 or V n (S n )), we prove that if an invariant c n−1 (α n−1 ) of the degenerate simplex is non-zero, then the vertices will be confined in a lower dimensional M n for any sufficiently small motion. This answers a question of the author. We also provide a simple geometric interpretation of c n−1 (α n−1 ) = 0 for the Euclidean case.