Abstract
In a recent work of Alessandro Fonda, a generalization of the parallelogram law in any dimension $N\geq 2$ was given by considering the ratio of the quadratic mean of the measures of the $(N-1)$-dimensional diagonals to the quadratic mean of the measures of the faces of a parallelotope. In this paper, we provide a further generalization considering not only $(N-1)$-dimensional diagonals and faces, but the $k$-dimensional ones for every $1\leq k\leq N-1$.
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