FURTHER GENERALIZATIONS OF THE PARALLELOGRAM

LAW
arXiv:1910.06645v1 [math.MG] 15 Oct 2019

ANTONIO M. OLLER-MARCÉN

Abstract. In recent work [2], a generalization of the parallelogram law in any

dimension N ≥ 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 ≤ k ≤ N − 1.

1. Introduction
If we consider the usual Euclidean space (Rn , k · k), the well-known identity
(1) ka + bk2 + ka − bk2 = 2(kak2 + kbk2 )
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) ka + b + ck2 + ka + b − ck2 + ka − b + ck2 + ka − b − ck2 = 4(kak2 + kbk2 + kck2 )
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
ka + bk2 + ka − bk2 (kak2 + kbk2 + kak2 + kbk2 )
(3) =2
2 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.
Theorem 1. Given linearly independent vectors a1 , . . . , aN ∈ Rn , it holds that
 2 2 

X  ^ ^ 

 (ai + aj ) ∧
ak + (ai − aj ) ∧ ak
=
i<j k6=i,j k6=i,j
N
X
= (N − 1) ck ∧ · · · ∧ aN k2 .
2 ka1 ∧ · · · ∧ a
k=1

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.
1
2 ANTONIO M. OLLER-MARCÉN

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.
Theorem 2. Let us consider an N -dimensional parallelotope and let 1 ≤ k ≤ N −1.
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 N − k + 1.
In fact, our generalization goes in the line of the work [3] but considering the
definition of diagonal face given in [2].

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 = {a1 , a2 , . . . , aN } ⊆ Rn . This means
that (N )
X
P= αi ai : αi ∈ [0, 1] .
i=1
Let us fix 1 ≤ k ≤ N − 1. Then, given k different vectors S = {ai1 , . . . , aik } ⊆ B,
we can consider the face generated by them:
( )
X
F (S) = αv v : αv ∈ [0, 1] .
v∈S

This face can now be translated by one or more of the remaining vectors thus
obtaining a face
 
X X 
F I (S) = αv av + αw w ∈ P : αw ∈ {0, 1} ,
 
v∈S w∈B\S
N −k
where I = (αv )v6∈S ∈ {0, 1} . 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 2N −k N k k-dimensional faces.
Moreover, it is clear that all the 2N −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 = {ai1 , . . . , aiN −k+1 } ⊆ B
and let T = T1 ∪ T2 be any partition. Then, the following set
 
 X X X 
D(T1 , T2 ) = α v + (1 − α) v+ αw w : α, αw ∈ [0, 1] .
 
v∈T1 v∈T2 w∈B\T

is called the k-dimensional diagonal associated to (T , T1 , T2 ). Clearly each choice of

a set T ⊆ B and a partition of T leads
to a different diagonal. Thus, it readily fol-
N
lows that P has exactly 2N −k N −k+1 different k-dimensional diagonals. Moreover,
if we define the vector X X
V (T1 , T2 ) = v− v,
v∈T1 v∈T2
 FURTHER GENERALIZATIONS OF THE PARALLELOGRAM LAW 3

we have that
 
 X X 
D(T1 , T2 ) = αV (T1 , T2 ) + v+ αw w : α, αw ∈ [0, 1]
 
v∈T2 w∈B\T

and, consequently, it is clear that the diagonal D(T1 , T2 ) is just a translation of the
set generated by {V (T1 , T2 ), w : w ∈ B \ T } and, hence, it is congruent to it.

3. 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 = {a1 , a2 , . . . , aN } ⊆ Rn . 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 = {ai1 , . . . , aik } ⊆ B, the k-
dimensional measure of the face F (S) is
kai1 ∧ · · · ∧ aik k. In the previous section
we have seen that P has exactly 2N −k N k k-dimensional faces and, moreover, that
there are exactly 2N −k copies of each face F (S). Consequently, the quadratic mean
of the k-dimensional measures of the k-dimensional faces of P is:
X
2N −k kai1 ∧ · · · ∧ aik k2
(4)   .
N −k N
2
k
Now we have to compute the quadratic mean of the k-dimensional measures of

N
the k-dimensional diagonals of P. First of all, recall that P has exactly 2N −k N −k+1
different k-dimensional diagonals. Each of them is the translation of the set gen-
erated by {V (T1 , T2 ), w : w ∈ B \ T } for exactly one choice of (T , T1 , T2 ). The k
^

dimensional measure of this latter set is V (T ,
1 2T ) ∧ w . Consequently, the

w∈B\T
quadratic mean of the k-dimensional measures of the k-dimensional diagonals of P
is:
2

X ^
V (T1 , T2 ) ∧ w

T ,T1 ,T2 w∈B\T
(5)   .
N
2N −k
N −k+1
Now, using the bilinearity of the scalar product and taking into account the defini-
tion of V (T1 , T2 ), it can be easily seen that when we vary (T , T1 , T2 ), we get the term
kai1 ∧· · ·∧aik k2 exactly 2N −K k times for every possible choice of {ai1 , . . . , aik } ⊆ B.
This implies that the quadratic mean of the k-dimensional measures of the k-
dimensional diagonals of P (5) can in fact be written as:
X
2N −k k kai1 ∧ · · · ∧ aik k2
(6)   .
N −k N
2
N −k+1
4 ANTONIO M. OLLER-MARCÉN

Finally, in order to obtain Theorem 2 it is enough to divide (6) by (4):

(6) k N k
= N

= N − k + 1.
(4) N −k+1

References
[1] Eeciolu, . Parallelogram-law-type identities. Linear Algebra Appl., 225:1–12, 1995.
[2] Fonda, A. A generalization of the parallelogram law to higher dimensions. Ars Mathematica
Contemporanea, 16(2):411–417, 2019.
[3] A generalized parallelogram law Nash, A. Amer. Math. Monthly, 110:52–57, 2003.
[4] Penico, A.J.; Stanojevi, .V. An integral analogue to parallelogram law. Proc. Amer. Math.
Soc., 79(3):427-430, 1980.

Centro Universitario de la Defensa de Zaragoza, Ctra. Huesca s/n, 50090 Zaragoza,

Spain
E-mail address: oller@unizar.es