A simple sum for simplices. (English) Zbl 1518.51016
Generalizing to \(n\) dimensions the result proved in [C. Aebi and G. Cairns, Coll. Math. J. 53, No. 5, 392–393 (2022; Zbl 1521.97005)] for \(n=2\), the authors prove by means of multilinear algebra that:
If \({\mathbf A}_0, \ldots, {\mathbf A}_{n+1}\) are \(n+2\) points in \({\mathbb R}^n\), understood as column vectors, and if \(\Delta_i\) denotes the determinant of the matrix \(M_i=|{\mathbf A}_{i+2}-{\mathbf A}_{i+1}| {\mathbf A}_{i+3}-{\mathbf A}_{i+1}| \ldots |{\mathbf A}_{i+n+1}-{\mathbf A}_{i+1}|\) (for \(i=0,\ldots, n+1\), where the indices are computed modulo \(n + 2\)), then \[ \sum_{i=0}^{n+1} (-1)^{i(n+1)}\Delta_i{\mathbf A}_i=0 \] The geometric meaning of \(\Delta_i\) is the following: the convex hull of \(n + 1\) points in \({\mathbb R}^n\) is a (possibly degenerate) \(n\)-simplex; \(\Delta_i\) the signed volume of the \(n\)-simplex defined by the \(n+1\) points among \({\mathbf A}_0, \ldots, {\mathbf A}_{n+1}\) that are different from \({\mathbf A}_i\).
If \({\mathbf A}_0, \ldots, {\mathbf A}_{n+1}\) are \(n+2\) points in \({\mathbb R}^n\), understood as column vectors, and if \(\Delta_i\) denotes the determinant of the matrix \(M_i=|{\mathbf A}_{i+2}-{\mathbf A}_{i+1}| {\mathbf A}_{i+3}-{\mathbf A}_{i+1}| \ldots |{\mathbf A}_{i+n+1}-{\mathbf A}_{i+1}|\) (for \(i=0,\ldots, n+1\), where the indices are computed modulo \(n + 2\)), then \[ \sum_{i=0}^{n+1} (-1)^{i(n+1)}\Delta_i{\mathbf A}_i=0 \] The geometric meaning of \(\Delta_i\) is the following: the convex hull of \(n + 1\) points in \({\mathbb R}^n\) is a (possibly degenerate) \(n\)-simplex; \(\Delta_i\) the signed volume of the \(n\)-simplex defined by the \(n+1\) points among \({\mathbf A}_0, \ldots, {\mathbf A}_{n+1}\) that are different from \({\mathbf A}_i\).
Reviewer: Victor V. Pambuccian (Glendale)
MSC:
| 51M20 | Polyhedra and polytopes; regular figures, division of spaces |
| 51M25 | Length, area and volume in real or complex geometry |
| 51N20 | Euclidean analytic geometry |
Citations:
Zbl 1521.97005References:
| [1] | C. Aebi and G. Cairns. A vector identity for quadrilaterals. To appear in College Math. J. Preprint available at arXiv:2106.11860. · Zbl 1521.97005 |
| [2] | Keith Conrad. Generating sets. Notes available at https://kconrad.math.uconn.edu/blurbs/grouptheory/genset.pdf. |
| [3] | Serge Lang. Algebra, revised third edition, Graduate Texts in Mathematics 211. Springer, 2002. |
| [4] | Loring, WTu, An Introduction to Manifolds (2011), Springer: Universitext, Springer · Zbl 1200.58001 |
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