[Paper Review] Reverses of the Triangle Inequality in Banach Spaces

Sever S Dragomir|arXiv (Cornell University)|May 25, 2005

Mathematics and Applications14 references21 citations

TL;DR

This paper presents new multiplicative and additive reverse inequalities for the triangle inequality in Banach and Hilbert spaces, extending classical results by Diaz-Metcalf and Petrovich. It establishes sharp bounds using angular constraints on vectors or functions, with key contributions in complex numbers, Hilbert spaces, and Bochner-integrable functions, including explicit conditions for equality and applications to polynomial root localization.

ABSTRACT

Recent reverses for the discrete generalised triangle inequality and its continuous version for vector-valued integrals in Banach spaces are surveyed. New results are also obtained. Particular instances of interest in Hilbert spaces and for complex numbers and functions are pointed out as well.

Motivation & Objective

• To survey and extend recent results on reverse triangle inequalities in Banach spaces, particularly multiplicative and additive forms.
• To address the lack of known reverse inequalities despite their importance in functional analysis and applications.
• To provide new sharp reverse inequalities for discrete and continuous cases in Hilbert spaces and for complex-valued functions.
• To establish conditions for equality in these reverse inequalities, enhancing their applicability.
• To apply the results to problems such as the location of roots of complex polynomials, as initiated by Marden and Wilf.

Proposed method

• Utilizes the Diaz-Metcalf framework by introducing a unit vector or functional with lower bounds on the real part of inner products.
• Applies angular constraints: for complex numbers and functions, assumes arguments lie within a sector of angle 2θ, leading to a cosine factor in the reverse inequality.
• Derives multiplicative reverses via norm and inner product estimates, e.g., r∑‖xᵢ‖ ≤ ‖∑xᵢ‖, with r related to the angle of vectors.
• Derives additive reverses by bounding the difference between the sum of norms and the norm of the sum, using a non-negative function k(t) as a slack term.
• Applies the method to Bochner integrable functions with values in Hilbert spaces, using vector-valued integral estimates.
• Extends results to Lᵖ norms on complex-valued functions by using ℓᵖ-type norms on real and imaginary parts, with appropriate normalization of coefficients.

Experimental results

Research questions

• RQ1 What are the sharp multiplicative and additive reverse inequalities for the triangle inequality in Banach spaces, and under what conditions do they hold?
• RQ2 How can the classical reverse triangle inequality for complex numbers (Petrovich-Diaz-Metcalf) be generalized to vector-valued and function-valued settings?
• RQ3 What are the necessary and sufficient conditions for equality in these reverse inequalities?
• RQ4 How can these reverse inequalities be applied to estimate the location of roots of complex polynomials?
• RQ5 What are the optimal constants in the reverse inequalities for Lᵖ norms on complex-valued functions?

Key findings

• For complex numbers with arguments in [a−θ, a+θ], the inequality cosθ∑|zₖ| ≤ |∑zₖ| holds, with equality if all zₖ are parallel and in the direction of e^{ia}.
• In Hilbert spaces, if vectors xᵢ satisfy Re⟨xᵢ,a⟩/‖xᵢ‖ ≥ r > 0 for a unit vector a, then r∑‖xᵢ‖ ≤ ‖∑xᵢ‖, with equality iff ∑xᵢ = r(∑‖xᵢ‖)a.
• For Bochner integrable functions f with values in a Hilbert space, if ‖f(t)‖ ≤ K Re⟨f(t),e⟩ a.e. for a unit vector e, then ∫‖f(t)‖dt ≤ K‖∫f(t)dt‖, with equality iff ∫f(t)dt = (1/K)(∫‖f(t)‖dt)e.
• For complex-valued functions, if |f(t)| ≤ φRe f(t) − ψIm f(t) + k(t) a.e. with φ² + ψ² = 1, then ∫|f(t)|dt − |∫f(t)dt| ≤ ∫k(t)dt, providing an additive reverse.
• For ℓᵖ norms, if [ |Re f(t)|²ᵖ + |Im f(t)|²ᵖ ]^{1/2p} ≤ φRe f(t) − ψIm f(t) + k(t) a.e. with φ² + ψ² = 2^{1/2p − 1/2}, then ∫[⋅]^{1/2p}dt − [ |∫Re f|²ᵖ + |∫Im f|²ᵖ ]^{1/2p} ≤ ∫k(t)dt.
• The results generalize known inequalities and provide a unified framework for reverse triangle inequalities in various settings, including complex numbers, Hilbert spaces, and vector-valued integrals.