Quadruple Inequalities: Between Cauchy-Schwarz and Triangle

TL;DR

This paper introduces a family of inequalities called quadruple inequalities that interpolate between the Cauchy-Schwarz and triangle inequalities, applicable in various metric spaces including Euclidean and CAT(0) spaces.

Contribution

It establishes a new class of inequalities derived from convex functions with concave derivatives, generalizing classical inequalities and introducing the quadruple constant to measure distortion.

Findings

Quadruple inequalities hold in metric spaces satisfying a metric Cauchy-Schwarz inequality.

A new quadruple constant quantifies the deviation from Cauchy-Schwarz.

An inner product space version generalizes the parallelogram law.

Abstract

We prove a set of inequalities that interpolate the Cauchy-Schwarz inequality and the triangle inequality. Every nondecreasing, convex function with a concave derivative induces such an inequality. They hold in any metric space that satisfies a metric version of the Cauchy-Schwarz inequality, including all CAT(0) spaces and, in particular, all Euclidean spaces. Because these inequalities establish relations between the six distances of four points, we call them quadruple inequalities. In this context, we introduce the quadruple constant - a real number that quantifies the distortion of the Cauchy-Schwarz inequality by a given function. Additionally, for inner product spaces, we prove an alternative, more symmetric version of the quadruple inequalities, which generalizes the parallelogram law.