Two Hornich-Hlawka-type and Gram matrix-based inequalities

TL;DR

This paper introduces two new inequalities in real inner product spaces, strengthening classical results and characterizing equality cases through Gram matrix conditions, with implications for the hierarchy of inequalities and inner product relations.

Contribution

It establishes a strengthened Hornich-Hlawka inequality with complete equality characterization and derives a parametric inequality from Gram matrix positivity, extending classical inequalities.

Findings

Complete characterization of equality cases in the Hornich-Hlawka inequality.

Derivation of a parametric inequality from Gram matrix positive semidefiniteness.

Recovery and strengthening of the classical Cauchy-Schwarz inequality.

Abstract

We establish two inequalities in real inner product spaces. The first is a multiplicative strengthening of the classical Hornich-Hlawka inequality: for all vectors $x, y, z$ in a real inner product space $H$ \[ \|x\|\,\|y\| + \|z\|\,\|x+y+z\| \;\geq\; \|x+z\|\,\|y+z\|. \] We provide a complete characterization of the equality cases in terms of the linear dependence of $x,y,z$, and explicit conditions on their Gram matrix, showing in particular that equality occurs only in flat (at most two-dimensional) configurations. We also show that this inequality implies the classical Hornich-Hlawka inequality, thereby establishing a strict hierarchy between the two. The second result is a parametric inequality derived from the positive semidefiniteness of Gram matrices: for all $x,y,z \in H$ and $\alpha, \beta, \gamma \in \mathbb{R}$, \[ \alpha^2\|x\|^2\langle y,z\rangle^2 + \beta^2\|y\|^2\langle x,z\rangle^2 + \gamma^2\|z\|^2\langle x,y\rangle^2 + 2(\alpha\beta + \alpha\gamma + \beta\gamma)\langle x,y\rangle\langle x,z\rangle\langle y,z\rangle \;\geq\; 0. \] Optimizing over the parameters yields sharp inequalities relating the pairwise inner products and norms of three vectors, which can be viewed as reverse inequalities to the Gram determinant inequality $\det G \geq 0$. As a special case, this recovers and strengthens the classical Cauchy-Schwarz inequality.