Further generalizations of the parallelogram law

TL;DR

This paper extends the generalization of the parallelogram law to include ratios involving all k-dimensional diagonals and faces of parallelotopes in any dimension, broadening the geometric understanding of these shapes.

Contribution

It introduces a new generalization considering all k-dimensional diagonals and faces, expanding previous work limited to (N-1)-dimensional elements.

Findings

Generalized parallelogram law for all k-dimensional elements

Applicable in any dimension N ≥ 2

Provides new geometric inequalities

Abstract

In recent work by 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 $1\leq k\leq N-1$.