Mathematics – Metric Geometry
Scientific paper
2007-02-21
Mathematics
Metric Geometry
23 pages, 4 figures, corrected typos and minor mistakes, simplified Thm 3.1 & 3.2 proof, 1 fig removed
Scientific paper
Given a degenerate $(n+1)$-simplex in a Euclidean space $R^n$ and a $k$ with $1\leq k\leq n$, we separate all its $k$-faces into 2 groups by following certain rules. The vertices are allowed to have continuous motion in $R^{n}$ while the volume of the $k$-faces in the 1st group can not increase (these faces are called ``$k$-cables''), and the volume of the $k$-faces in the 2nd group can not decrease (``$k$-struts''). Assuming some smoothness property, we prove that all the volumes of the $k$-faces will be preserved for any sufficiently small motion. When the vertices are allowed to move in $R^{n+1}$, we derive a $n$-degree ``characteristic polynomial'' for the points configuration, and prove that this property still holds if the $(k-1)$-th coefficient of the polynomial has the desired sign. We also show that all the roots of the polynomial are real. We generalize the above results to spherical space $S^n$ and hyperbolic space $H^n$ as well.
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