# A Sharp Norm Inequality and Buzano's Inequality via Determinants

arXiv: https://arxiv.org/abs/2604.01525 · HTML: https://mlsanity.com/paper/2604.01525/

## Key facts

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- First posted on arXiv 2 Apr 2026.

## Abstract (arXiv)

We give a short linear-algebraic proof of the inequality  |x|1,|x|_∞ ≤ frac1+√n2,|x|22,  valid for every x∈ℝn. This inequality relates three fundamental norms on finite-dimensional spaces and has applications in optimization and numerical analysis. Our proof exploits the determinantal structure of a parametrized family of quadratic forms, and we show the constant (1+√n)/2 is optimal. The inequality is a special case of Buzano's inequality, and the same method also proves Buzano's inequality for real vectors.