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

Further generalizations of the parallelogram law