A Determinantal Approach to a Sharp $\ell^1-\ell^\infty-\ell^2$ Norm Inequality
This provides a foundational mathematical result with applications in optimization and numerical analysis, but it is incremental as it offers a new proof method rather than a new inequality.
The paper tackles the problem of proving a sharp inequality relating the ℓ¹, ℓ∞, and ℓ² norms for vectors in ℝᵖ, showing that ‖x‖₁‖x‖∞ ≤ (1+√p)/2 ‖x‖₂² with an optimal constant.
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.