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

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. The second result is a parametric inequality derived from the positive semidefiniteness of Gram matrices: for all $x,y,z\in H$ and $\alpha ,\beta ,\gamma \in \mathbb{R}$, \[ α^2\|x\|^2\langle y,z\rangle^2 + β^2\|y\|^2\langle x,z\rangle^2 + γ^2\|z\|^2\langle x,y\rangle^2 + 2(αβ+ αγ+ βγ)\langle x,y\rangle\langle x,z\rangle\langle y,z\rangle \;\geq\; 0 \], which, upon optimization over the parameters yields sharp reverse inequalities to the Gram determinant inequality $\det G\ge 0$. As a special case, this recovers and strengthens the classical Cauchy-Schwarz inequality.

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