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This paper presents a concise, linear-algebraic proof for a sharp inequality relating the $\ell^1$, $\ell^\infty$, and $\ell^2$ norms of vectors in $\mathbb{R}^p$. The proof leverages the determinantal structure of a parametrized family of quadratic forms. The authors demonstrate that the constant $(1+\sqrt{p})/2$ in the inequality $\|x\|_1\,\|x\|_\infty \le \frac{1+\sqrt{p}}{2}\,\|x\|_2^2$ is optimal, strengthening its applicability in optimization and numerical analysis.
A surprisingly simple determinantal argument yields the tightest possible bound relating $\ell^1$, $\ell^\infty$, and $\ell^2$ norms.
We give a short linear--algebraic proof of the inequality \[ \|x\|_1\,\|x\|_\infty \le \frac{1+\sqrt{p}}{2}\,\|x\|_2^2, \] valid for every \(x\in\mathbb{R}^p\). 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 $(1+\sqrt{p})/2$ is optimal.