[Paper Review] Characterization of a generalized triangle inequality in normed spaces

Farzad Dadipour, Mohammad Sal Moslehian+2 more|arXiv (Cornell University)|Sep 8, 2011

Mathematical Inequalities and Applications11 references3 citations

TL;DR

This paper characterizes all $ n $-tuples $ (\mu_1, \dots, \mu_n) \in \mathbb{R}^n $ for which the generalized triangle inequality of the second type, $ \|x_1 + \cdots + x_n\|^p \leq \sum_{i=1}^n \frac{\|x_i\|^p}{\mu_i} $, holds for all $ x_i $ in a normed space $ X $, and also identifies when the reverse inequality holds. The key result provides explicit necessary and sufficient conditions on the $ \mu_i $, involving $ p $-norm-like inequalities and envelope theory, distinguishing cases for $ p > 1 $ and $ 0 < p \leq 1 $.

ABSTRACT

For a normed linear space $(X,|\cdot|)$ and $p&gt;0$ we characterize all $n$-tuples $(μ_1,...,μ_n)\in\mathbb{R}^{n}$ for which the generalized triangle inequality of the second type $$\|x_1+...+x_n\|^p\leq\frac{|x_1|^p}{μ_1}+...+\frac{|x_n|^p}{μ_n}$$ holds for any $x_1,...,x_n\in X$. We also characterize $(μ_1,...,μ_n)\in\mathbb{R}^{n}$ for which the reverse of the inequality above holds.

Motivation & Objective

• Characterize all $ n $-tuples $ (\mu_1, \dots, \mu_n) \in \mathbb{R}^n $ such that the generalized triangle inequality of the second type holds for all $ x_1, \dots, x_n \in X $ in a normed space $ (X, \|\cdot\|) $.
• Establish necessary and sufficient conditions on $ \mu_i $ for the reverse inequality to hold.
• Investigate the structure of the set of $ \mu $-tuples for different ranges of $ p > 0 $, particularly distinguishing $ p > 1 $ and $ 0 < p \leq 1 $.
• Extend known results on triangle inequalities in Hilbert and normed spaces to the $ n $-variable, $ p $-th power generalized form.
• Use envelope theory and geometric analysis of level sets to derive exact characterizations of the feasible $ \mu $-regions.

Proposed method

• The paper employs envelope theory for families of surfaces defined by $ a_1 s_1^p + \cdots + a_n s_n^p = 1 $, where $ s_i \geq 0 $, $ \sum s_i = 1 $, to characterize the boundary of the feasible region for $ \mu $-tuples.
• Key lemmas define the envelope function $ h_p(a_1, \dots, a_{n-1}) = \left(1 - \sum_{i=1}^{n-1} a_i^{1/(1-p)} \right)^{1-p} $, which determines the critical threshold for $ \mu_n $ in terms of the other $ \mu_i $.
• By analyzing the intersection of superlevel sets $ \Delta_p(s_1, \dots, s_n) $, the paper derives the condition $ a_n \geq h_p(a_1, \dots, a_{n-1}) $, which translates into constraints on the $ \mu_i $.
• The proof uses substitution and normalization, reducing the inequality to a form where the parameter $ \lambda = \mu a^2 + \nu b^2 $ is absorbed via scaling, generalizing known Euler-Lagrange-type identities.
• Geometric and analytic techniques are applied to the set $ \Omega $ of normalized norm ratios, showing that feasibility depends on whether certain scaled vectors lie in the dual set $ D_p(\Omega) $.
• Critical inequalities are derived via substitution: for $ p > 1 $, $ \mu_j^{1/(p-1)} \geq 1 + \sum_{i \neq j} |\mu_i|^{1/(p-1)} $, and for $ 0 < p \leq 1 $, $ \mu_j \geq \max_{i \neq j} \{1, |\mu_i|\} $, with $ \mu_j > 0 $, others negative.

Experimental results

Research questions

• RQ1 What are the necessary and sufficient conditions on $ (\mu_1, \dots, \mu_n) \in \mathbb{R}^n $ such that $ \|x_1 + \cdots + x_n\|^p \leq \sum_{i=1}^n \frac{\|x_i\|^p}{\mu_i} $ holds for all $ x_i \in X $ in a normed space $ X $ and for all $ p > 0 $?
• RQ2 For which $ n $-tuples $ (\mu_1, \dots, \mu_n) $ does the reverse inequality $ \|x_1 + \cdots + x_n\|^p \geq \sum_{i=1}^n \frac{\|x_i\|^p}{\mu_i} $ hold universally in a normed space?
• RQ3 How do the feasible $ \mu $-regions differ between the cases $ p > 1 $ and $ 0 < p \leq 1 $, and what structural constraints define these regions?
• RQ4 What role does the envelope of a family of surfaces defined by $ a_1 s_1^p + \cdots + a_n s_n^p = 1 $ play in characterizing the boundary of the feasible $ \mu $-set?
• RQ5 Can the generalized triangle inequality of the second type be reduced to a known identity or inequality in special cases, such as $ n = 2 $ or Hilbert spaces?

Key findings

• For $ p > 1 $, the set of $ \mu $-tuples for which the generalized triangle inequality holds is $ F(p) \cap (-F(p)) $, where $ F(p) $ consists of $ \mu $-tuples with all $ \mu_i > 0 $ and satisfying $ \sum_{i=1}^n |\mu_i|^{1/(p-1)} \leq 1 $.
• When $ p > 1 $, the reverse inequality holds if and only if all $ \mu_i < 0 $ and $ \sum_{i=1}^n |\mu_i|^{1/(p-1)} \leq 1 $, forming the set $ -F(p) $.
• For $ 0 < p \leq 1 $, the feasible $ \mu $-tuples for the forward inequality are exactly those with all $ \mu_i \in (0,1] $, and for the reverse inequality, all $ \mu_i \in [-1,0) $, forming $ (0,1]^n \cup [-1,0)^n $.
• The critical condition for $ p > 1 $ is $ \mu_j^{1/(p-1)} \geq 1 + \sum_{i \neq j} |\mu_i|^{1/(p-1)} $, which arises from the envelope condition when one $ \mu_j > 0 $ and others are negative.
• For $ 0 < p \leq 1 $, the condition $ \mu_j \geq \max_{i \neq j} \{1, |\mu_i|\} $ is necessary and sufficient when $ \mu_j > 0 $ and all other $ \mu_i < 0 $, characterizing the feasible set $ G(p; n-1) $.
• The full set $ H(p) $ of $ \mu $-tuples satisfying the inequality in absolute value form is $ H(p) = (F(p) \cap -G(p)) \cup (-F(p) \cap G(p)) $, which simplifies to the stated unions for $ p > 1 $ and $ 0 < p \leq 1 $.