Pythagorean theorem, multi-dimensional

Consider the -dimensional space (with the usual metric and measure). Let be a point on the th coordinate axis and let be the origin. Let be the -dimensional volume of the -dimensional simplex and let be the -dimensional volume of the -dimensional simplex . Then .

For other and further generalizations of the classical Pythagoras theorem, see [a2] and the references therein.

References

[a1]	Etsua Yoshinaga, Shigeo Akiba, "Very simple proofs of the generalized Pythagorean theorem" Sci. Reports Yokohama National Univ. Sect. I , 42 (1995) pp. 45–46
[a2]	D.R. Conant, W.A. Beyer, "Generalized Pythagorean theorem" Amer. Math. Monthly , 81 (1974) pp. 262–265