In a recent work of Alessandro Fonda, 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

Oller-Marcen, A.M.

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Repositorio Universidad de Zaragoza

arXiv:1910.06645v1  [math.MG]  15 Oct 2019FURTHER GENERALIZATIONS OF THE PARALLELOGRAMLAWANTONIO M. OLLER-MARCÉNAbstract. In recent work [2], a generalization of the parallelogram law in anydimension N ≥ 2 was given by considering the ratio of the quadratic mean ofthe measures of the N − 1-dimensional diagonals to the quadratic mean of themeasures of the faces of a parallelotope. In this paper, we provide a furthergeneralization considering not only (N − 1)-dimensional diagonals and faces,but the k-dimensional ones for every 1 ≤ k ≤ N − 1.1. IntroductionIf we consider the usual Euclidean space (Rn, ‖ · ‖), the well-known identity(1) ‖a+ b‖2 + ‖a− b‖2 = 2(‖a‖2 + ‖b‖2)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) ‖a+ b+ c‖2+ ‖a+ b− c‖2+ ‖a− b+ c‖2+ ‖a− b− c‖2 = 4(‖a‖2+ ‖b‖2+ ‖c‖2)with subsequent analogue identities arising inductively. There are, in fact, manyworks devoted to provide generalizations of (1) in many different contexts [1, 3, 4]Note that if we rewrite (1) as(3)‖a+ b‖2 + ‖a− b‖22= 2(‖a‖2 + ‖b‖2 + ‖a‖2 + ‖b‖2)4it just means that in any parallelogram, the ratio of the quadratic mean of thelengths 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 provedthe following interesting generalization.Theorem 1. Given linearly independent vectors a1, . . . , aN ∈ Rn, it holds that∑i<j∥∥∥∥∥∥(ai + aj) ∧∧k 6=i,jak∥∥∥∥∥∥2+∥∥∥∥∥∥(ai − aj) ∧∧k 6=i,jak∥∥∥∥∥∥2 == (N − 1)N∑k=12 ‖a1 ∧ · · · ∧ âk ∧ · · · ∧ aN‖2 .In other words, for any N -dimensional parallelotope, the ratio of the quadratic meanof the (N − 1)-dimensional measures of its diagonals to the quadratic mean of the(N − 1)-dimensional measures of its faces is equal to√2.12 ANTONIO M. OLLER-MARCÉNIn this work we extend this result considering the faces of dimension k for every1 ≤ 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-dimensionaldiagonals to the quadratic mean of the k-dimensional measures of its k-dimensionalfaces is equal to√N − k + 1.In fact, our generalization goes in the line of the work [3] but considering thedefinition of diagonal face given in [2].2. Notation and preliminariesIn this section we are going to introduce some notation and to present some basicfacts that will be useful in the sequel. Let us consider a parallelotope P generatedby a family of linearly independent vectors B = {a1, a2, . . . , aN} ⊆ Rn. This meansthatP ={N∑i=1αiai : αi ∈ [0, 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:F(S) ={∑v∈Sαvv : αv ∈ [0, 1]}.This face can now be translated by one or more of the remaining vectors thusobtaining a faceFI(S) =∑v∈Sαvav +∑w∈B\Sαww ∈ P : αw ∈ {0, 1} ,where I = (αv)v 6∈S ∈ {0, 1}N−k. Since each choice of a set S ⊆ B and a vectorI ∈ {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(Nk)k-dimensional faces.Moreover, it is clear that all the 2N−k different faces FI(S) are congruent to theset generated by S, F(S).Now, we focus on the k-dimensional diagonals which will be defined following theideas in [2]. Let us consider N − k+1 different vectors T = {ai1 , . . . , aiN−k+1} ⊆ Band let T = T1 ∪ T2 be any partition. Then, the following setD(T1, T2) =α∑v∈T1v + (1− α)∑v∈T2v +∑w∈B\Tαww : α, αw ∈ [0, 1] .is called the k-dimensional diagonal associated to (T , T1, T2). Clearly each choice ofa set T ⊆ B and a partition of T leads to a different diagonal. Thus, it readily fol-lows that P has exactly 2N−k(NN−k+1)different k-dimensional diagonals. Moreover,if we define the vectorV (T1, T2) =∑v∈T1v −∑v∈T2v,FURTHER GENERALIZATIONS OF THE PARALLELOGRAM LAW 3we have thatD(T1, T2) =αV (T1, T2) +∑v∈T2v +∑w∈B\Tαww : α, αw ∈ [0, 1]and, consequently, it is clear that the diagonal D(T1, T2) is just a translation of theset generated by {V (T1, T2), w : w ∈ B \ T } and, hence, it is congruent to it.3. Proof of Theorem 2After introducing the notation and the main objects involved in thie work, weare 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 firstcompute the quadratic mean of the k-dimensional measures of its k-dimensionalfaces. To do so, we first note that, for every S = {ai1 , . . . , aik} ⊆ B, the k-dimensional measure of the face F(S) is ‖ai1 ∧ · · · ∧ aik‖. In the previous sectionwe have seen that P has exactly 2N−k(Nk)k-dimensional faces and, moreover, thatthere are exactly 2N−k copies of each face F(S). Consequently, the quadratic meanof the k-dimensional measures of the k-dimensional faces of P is:(4)2N−k∑‖ai1 ∧ · · · ∧ aik‖22N−k(Nk) .Now we have to compute the quadratic mean of the k-dimensional measures ofthe k-dimensional diagonals of P . First of all, recall that P has exactly 2N−k(NN−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 kdimensional measure of this latter set is∥∥∥∥∥∥V (T1, T2) ∧∧w∈B\Tw∥∥∥∥∥∥. Consequently, thequadratic mean of the k-dimensional measures of the k-dimensional diagonals of Pis:(5)∑T ,T1,T2∥∥∥∥∥∥V (T1, T2) ∧∧w∈B\Tw∥∥∥∥∥∥22N−k(NN − 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‖ai1∧· · ·∧aik‖2 exactly 2N−Kk 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:(6)2N−kk∑‖ai1 ∧ · · · ∧ aik‖22N−k(NN − k + 1) .4 ANTONIO M. OLLER-MARCÉNFinally, in order to obtain Theorem 2 it is enough to divide (6) by (4):(6)(4)=k(Nk)(NN−k+1) = 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 MathematicaContemporanea, 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,SpainE-mail address: oller@unizar.es

Further generalizations of the parallelogram law

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Further generalizations of the parallelogram law

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