| Abstrakt: |
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 ∑ X 1 V n (F i) = ∑ X 2 V n (F i) , except for a special spherical case where X 2 is the empty set and ∑ X 1 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 ∑ X 1 V n (F i) - ∑ X 2 V n (F i) is preserved as a constant (0 or V n (S n) ), we prove that if a stress invariant c n - 1 (α n - 1) of the degenerate simplex is non-zero, then the vertices will be confined to a lower dimensional M n for any sufficiently small motion. This answers a question of the author and we also show that in the Euclidean case, c n - 1 (α n - 1) = 0 is equivalent to the vertices of a dual degenerate (n + 1) -simplex lying on an (n - 1) -sphere in R n . [ABSTRACT FROM AUTHOR] |
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