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

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 $H$ $$\mathrm{\parallel }x\mathrm{\parallel }\mathrm{\parallel }y\mathrm{\parallel }+\mathrm{\parallel }z\mathrm{\parallel }\mathrm{\parallel }x+y+z\mathrm{\parallel }\ge \mathrm{\parallel }x+z\mathrm{\parallel }\mathrm{\parallel }y+z\mathrm{\parallel }\mathrm{.}$$ 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 $\alpha ,\beta ,\gamma \in \mathbb{R}$, $${\alpha }^2\mathrm{\parallel }x{\mathrm{\parallel }}^2⟨y,z{⟩}^2+{\beta }^2\mathrm{\parallel }y{\mathrm{\parallel }}^2⟨x,z{⟩}^2+{\gamma }^2\mathrm{\parallel }z{\mathrm{\parallel }}^2⟨x,y{⟩}^2+2(\alpha \beta +\alpha \gamma +\beta \gamma )⟨x,y⟩⟨x,z⟩⟨y,z⟩\ge 0.$$ 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 $\det G\ge 0$. As a special case, this recovers and strengthens the classical Cauchy-Schwarz inequality.