A Sharp Norm Inequality and Buzano's Inequality via Determinants

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.