Quadruple Inequalities: Between Cauchy-Schwarz and Triangle
Christof Schötz
TL;DR
This work introduces quadruple inequalities that interpolate between the triangle inequality and Cauchy–Schwarz in metric spaces, valid whenever a base four-point (quadruple) inequality holds. It defines quadruple transformations $\tau$ and a quadruple constant $L_\tau^*$ to quantify how much a transformed inequality can distort the CS-type bound, and proves that all nondecreasing convex functions with concave derivatives lie in the transformation set $\mathcal{T}$ with $L_\tau^*\in[1,2]$, with equality $L_\tau^*=1$ only in trivial cases. In inner product spaces, the authors obtain a stronger, parallelogram-law–like bound for $\tau\in\mathcal{S}_0$, linking distance sums to transformed distances in a symmetric fashion. The results connect to CAT$(0)$ geometry, four-point cosq conditions, and Ptolemy-type inequalities, and have implications for Fréchet means and robust statistics in metric spaces. Overall, the paper provides a unified, function-parameterized framework to generalize CS and triangle inequalities and to study the resulting geometric and statistical consequences.
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.