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A comprehensive Pythagorean theorem for all dimensions. (English)
Am. Math. Mon. 122, No. 2, 164-168 (2015).
It is shown that the squared $n$-dimensional volume of an $n$-dimensional parallelotope in $\mathbb R ^m$, multiplied by $m-n\choose k-n$, equals the sum of squared $n$-dimensional volumes of all projections of the polytope to $k$-dimensional coordinate subspaces. The proof is a simple application of the Cauchy-Binet determinantal formula.
Reviewer: P├ęter E. Frenkel (Budapest)
Classification: H65 G95 G45
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