Further generalizations of the parallelogram law

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 $1\leq k\leq N-1$.


Introduction
If we consider the usual Euclidean space (R n , • ), the well-known identity (1) a is called the parallelogram law.This identity can be extended to higher dimensions in several ways.For example, it is straightforward to see that (2) with subsequent analogue identities arising inductively.There are, in fact, many works devoted to provide generalizations of (1) in many different contexts [1,3,4] Note that if we rewrite (1) as 4 it just means that in any parallelogram, the ratio of the quadratic mean of the lengths of its diagonals to the quadratic mean of the lengths of its sides equals √ 2. With this interpretation in mind, Alessandro Fonda [2] has recently proved the following interesting generalization.
In other words, for any N -dimensional parallelotope, the ratio of the quadratic mean of the (N − 1)-dimensional measures of its diagonals to the quadratic mean of the (N − 1)-dimensional measures of its faces is equal to √ 2.
In this work we extend this result considering the faces of dimension k for every 1 ≤ k ≤ N −1 and a suitable definition of k-dimensional diagonal of a parallelotope.Then, Theorem 1 will just be a particular case of our result for k = N − 1.Indeed, our result can be stated as follows.
The ratio of the quadratic mean of the k-dimensional measures of its k-dimensional diagonals to the quadratic mean of the k-dimensional measures of its k-dimensional faces is equal to In fact, our generalization goes in the line of the work [3] but considering the definition of diagonal face given in [2].

Notation and preliminaries
In this section we are going to introduce some notation and to present some basic facts that will be useful in the sequel.Let us consider a parallelotope P generated by a family of linearly independent vectors B = {a 1 , a 2 , . . ., a N } ⊆ R n .This means that we can consider the face generated by them: This face can now be translated by one or more of the remaining vectors thus obtaining a face where Since each choice of a set S ⊆ B and a vector I ∈ {0, 1} N −k leads to a different face and every face can be obtained in this way, it follows the well-known result that P has exactly 2 N −k N k k-dimensional faces.Moreover, it is clear that all the 2 N −k different faces F I (S) are congruent to the set generated by S, F (S). Now, we focus on the k-dimensional diagonals which will be defined following the ideas in [2].Let us consider N − k + 1 different vectors T = {a i1 , . . ., a i N −k+1 } ⊆ B and let T = T 1 ∪ T 2 be any partition.Then, the following set is called the k-dimensional diagonal associated to (T , T 1 , T 2 ).Clearly each choice of a set T ⊆ B and a partition of T leads to a different diagonal.Thus, it readily follows that P has exactly 2 N −k N N −k+1 different k-dimensional diagonals.Moreover, if we define the vector we have that and, consequently, it is clear that the diagonal D(T 1 , T 2 ) is just a translation of the set generated by {V (T 1 , T 2 ), w : w ∈ B \ T } and, hence, it is congruent to it.

Proof of Theorem 2
After introducing the notation and the main objects involved in thie work, we are now in the condition to proof the main result of the paper.
Let P be a parallelotope generated by B = {a 1 , a 2 , . . ., a N } ⊆ R n .We first compute the quadratic mean of the k-dimensional measures of its k-dimensional faces.To do so, we first note that, for every S = {a i1 , . . ., a i k } ⊆ B, the kdimensional measure of the face F (S) is a i1 ∧ • • • ∧ a i k .In the previous section we have seen that P has exactly 2 N −k N k k-dimensional faces and, moreover, that there are exactly 2 N −k copies of each face F (S). Consequently, the quadratic mean of the k-dimensional measures of the k-dimensional faces of P is: Now we have to compute the quadratic mean of the k-dimensional measures of the k-dimensional diagonals of P. First of all, recall that P has exactly 2 N −k N N −k+1 different k-dimensional diagonals.Each of them is the translation of the set generated by {V (T 1 , T 2 ), w : w ∈ B \ T } for exactly one choice of (T , T 1 , T 2 ).The k dimensional measure of this latter set is V (T 1 , T 2 ) ∧ w∈B\T w .Consequently, the quadratic mean of the k-dimensional measures of the k-dimensional diagonals of P is: (5) . Now, using the bilinearity of the scalar product and taking into account the definition of V (T 1 , T 2 ), it can be easily seen that when we vary (T , T 1 , T 2 ), we get the term a i1 ∧• • •∧a i k 2 exactly 2 N −K k times for every possible choice of {a i1 , . . ., a i k } ⊆ B. This implies that the quadratic mean of the k-dimensional measures of the kdimensional diagonals of P (5) can in fact be written as: .