Two Hornich-Hlawka-type and Gram matrix-based inequalities
Nizar El Idrissi, Hicham Zoubeir
TL;DR
This work investigates three-vector inequalities in inner product spaces by combining two strategies: a strengthened Hornich–Hlawka inequality that refines the classical $\|x\|\|y\| + \|z\|\|x+y+z\| \geq \|x+y\|\|y+z\|$ and a Gram-matrix framework that yields sharp, CS-type bounds. The stronger inequality, proved via a quartic form in Gram matrix entries and boundary analysis of the PSD cone, characterizes equality only in flat, linearly dependent configurations. The Gram-matrix approach shows that nonnegativity of the Gram matrix $G$ implies a parameterized inequality with $\alpha,\beta,\gamma$ that strengthens the Cauchy–Schwarz inequality and leads to determinant-type corollaries. Together, these results provide a unified, geometry-driven toolkit for generating and understanding three-vector inequalities in inner product spaces, with equality cases tied to degeneracies such as collinearity or planarity of the vectors.
Abstract
This paper deals with Hornich-Hlawka-type and Gram matrix-based inequalities in inner product spaces (IPS). On the one hand, the Hornich-Hlawka inequality, or Hlawka's inequality, or quadrilateral inequality, is an inequality concerning three vectors in a quadrilateral. Namely, it states that for all $x,y,z$, $\lVert x \rVert + \lVert y \rVert + \lVert z \rVert + \lVert x + y + z \rVert \geq \lVert x+y \rVert + \lVert x + z \rVert + \lVert y + z \rVert$. On the other hand, inequalities based on Gram matrices are very important because they generate generically all inequalities implying three vectors in an IPS. In the first part of our paper, we introduce a stronger version of Hornich-Hlawka inequality, namely, for all vectors $ x, y, z $, $ \|x\| \|y\| + \|z\| \|x + y + z\| \geq \|x + z\| \|y + z\|$. In the second part of our paper, we establish a sharp inequality based on Gram matrices that implies, in particular, the standard Cauchy-Schwarz inequality.