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A generalization of the parallelogram law to higher dimensions. (English) Zbl 1423.51009

It is a well-known and easy result that, for any two vector \(a\) and \(b\) it holds that \[ ||a+b||^2+||a-b||^2=2(||a||^2+||b||^2). \] This so-called parallelogram law is equivalent to: “For any parallelogram, the ratio of the quadratic mean of the lengths of its diagonals to the quadratic mean of the lengths of its sides is equal to \(\sqrt{2}\).”
In this paper, the author proves the same result for the case of \(N\)-dimensional parallelotopes.
The result is rather unsurprising and straightforward, since it is only based on the already known fact that the volume (understood as the \(M\)-dimensional measure) of a parallelotope defined by vectors \(v_1,\dots,v_M\in\mathbb{R}^N\) is just \(||v_1\wedge\cdots\wedge v_M||\).

MSC:

51M04 Elementary problems in Euclidean geometries
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References:

[1] M. Khosravi and M. D. Taylor, The wedge product and analytic geometry, Amer. Math. Monthly 115 (2008), 623-644, doi:10.1080/00029890.2008.11920573. · Zbl 1159.51007
[2] A. Nash, A generalized parallelogram law, Amer. Math. Monthly 110 (2003), 52-57, doi:10. 2307/3072345. · Zbl 1026.52007
[3] I. R. Shafarevich and A. O.
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