The triangle inequality for graded real vector spaces
Songpon Sriwongsa, Keng Wiboonton

TL;DR
This paper proves that a natural homogeneous norm on graded Lie algebras satisfies the triangle inequality, confirming a key property for these mathematical structures and answering a previously posed question.
Contribution
It establishes the triangle inequality for a natural candidate norm on graded Lie algebras of any length, advancing understanding of their geometric properties.
Findings
The candidate norm satisfies the triangle inequality.
The result applies to graded Lie algebras of any length.
Answers Moskowitz's question affirmatively.
Abstract
In this paper, we prove that a natural candidate for a homogeneous norm on a graded Lie algebra of any length satisfies the triangle inequality which answers Moskowitz's question.
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Taxonomy
TopicsMathematics and Applications · Optimization and Variational Analysis · Advanced Differential Equations and Dynamical Systems
The triangle inequality for graded real vector spaces of length
Songpon Sriwongsa and Keng Wiboonton
Songpon Sriwongsa
Department of Mathematics
Faculty of Science
King Mongkut’s University of Technology Thonburi (KMUTT)
Bangkok 10140, Thailand
[email protected], [email protected]
Keng Wiboonton
Department of Mathematics and Computer Science
Faculty of Science
Chulalongkorn University
Bangkok 10330, Thailand
Abstract.
In this paper, we prove that a natural candidate for a homogeneous norm on a graded Lie algebra of length 5 satisfies the triangle inequality which answers Moskowitz’s question in [2].
Key words and phrases:
Graded Lie algebra, subadditive homogeneous norm.
2010 Mathematics Subject Classification:
17B70, 22E25, 26D15
1. Introduction
In [1], M. Moskowitz extended the classical theorem of Minkowski on lattice points and convex bodies in to simply connected nilpotent Lie groups with a -structure whose Lie algebra admits a length 2 grading. For this purpose, the author proved the triangle inequality for a certain natural homogeneous norm of the Lie algebra associated with the grading of length 2. Later, in [2], the same author extended the homogeneous norms for which the triangle inequality holds to gradings of length 3 and 4. At the end of the paper [2], the author demonstrated that the method of his proof cannot be carried on to the case of length 5, so it was left as an open problem. However, the arguments in the proof contains some errors. The inequalities (6) and (12) in the proof of Theorem 1 in [2] are not correct for length or .
In this paper, we claim that for the grading of lengths 3, 4 and 5, the homogeneous norms defined as in [2] satisfy the the triangle inequality. For the sake of conciseness, the complete proof will be provided only for the most complicated case, the case of length 5. The gradings of length 3 and 4 can be done with the same manner. Our proof is quite elementary. The main tools in our proof are only Hölder’s inequality and Muirhead’s inequality. We refer the reader to [3] for the latter inequality.
A Lie algebra is said to be graded if there is a finite family of subspaces with satisfying for all (here, if ). The integer is its length. For a graded Lie algebra , we define, for ,
[TABLE]
Then is Lie algebra automorphism of . Moreover, if is a graded Lie algebra, then must be nilpotent (see [1] and [2]).
As in [1] and [2], we define a homogeneous norm on a graded Lie algebra as a function satisfying the following conditions.
- (i)
and is [math] only at [math]. 2. (ii)
for all . 3. (iii)
for all and . 4. (iv)
for all .
Any graded Lie algebra possesses a natural candidate for a homogeneous norm as follows: for , let
[TABLE]
where is the Euclidean norm on each (we may suppress the subscript for norms). This norm satisfies properties (i) and (ii). However, the the homogeneity property (iii) holds if and only if . In [1], it was shown that the subadditivity property (iv) (or the triangle inequality) holds when . Our main result is that the triangle inequality holds for and .
Theorem 1**.**
* for .*
2. Proof of the main theorem
We prove the main theorem for length only and one will see that the similar arguments can be used to prove for length and with slightly modification but easier.
Proof of the main theorem: Let and . Then
[TABLE]
By taking th powers, we need to prove that
[TABLE]
Expanding both sides by using the binomial theorem, applying the Schwartz inequality to all norms, and canceling yield
[TABLE]
where
[TABLE]
and
[TABLE]
Consider the middle terms of each summation, we have
[TABLE]
which, in fact, holds by Holder’s inequality (with ) since the largest coefficient dominates the others.
Next we show that
[TABLE]
We note that taking does not effect this inequality. By Hölder’s inequality, we have the two following inequalities:
[TABLE]
Thus, we only need to show that
[TABLE]
These two inequalities follow from Muirhead’s inequality because majorizes and majorizes . Hence, the equality (2.1) holds.
By the similar arguments (with some changes on exponents), we can prove another two inequalities:
[TABLE]
and
[TABLE]
It remains to show that . We can see that this is obvious by taking for all . ∎
Remark**.**
The above proof was done by using elementary tools including Hölder’s inequality and Muirhead’s inequality. We strongly believe that the proof can be extended to any length with the same manner. The crucial steps must be to match up all suitable terms from the left and the right of inequalities, so that Hölder’s inequality and Muirhead’s inequality can be used.
The reference list from the paper itself. Each links out to its DOI / PubMed record.
- 1[1] M. Moskowitz, An extension of Minkowski’s convex body theorem to certain simply connected nilpotent groups , Portugaliae Mathematica vol. 67 , no. 4 (2010), 541–546.
- 2[2] M. Moskowitz, The triangle inequality for graded real vector spaces of length 3 and 4 , Math. Inequal. Appl. 17 , no. 3 (2014), 1027–1030.
- 3[3] J. M. Steele, The Cauchy-Schwarz master class , Cambridge University Press, 2004.
