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- W1662470844 abstract "Given a degenerate $$(n+1)$$ -simplex in a d-dimensional space $$M^d$$ (Euclidean, spherical or hyperbolic space, and $$dge n$$ ), for each k, $$1le kle n$$ , 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 $$dge n+1$$ if an invariant $$c_{k-1}(alpha ^{k-1})$$ of the degenerate simplex has the desired sign. This answers a question posed by the author, and the proof relies on an invariant $$c_k(omega )$$ we discovered for any k-stress $$omega $$ on a cell complex in $$M^d$$ . We introduce a characteristic polynomial of the degenerate simplex by defining $$f(x)=sum _{i=0}^{n+1}(-1)^{i}c_i(alpha ^i)x^{n+1-i}$$ , and prove that the roots of f(x) are real for the Euclidean case. Some evidence suggests the same conjecture for the hyperbolic case." @default.
- W1662470844 created "2016-06-24" @default.
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- W1662470844 date "2017-12-05" @default.
- W1662470844 modified "2023-09-27" @default.
- W1662470844 title "Rigidity and Volume Preserving Deformation on Degenerate Simplices" @default.
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- W1662470844 doi "https://doi.org/10.1007/s00454-017-9956-x" @default.
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