This inequality is named after Hermann Minkowski (1864 - 1909). It is a generalization of the triangle inequality for $p=1.$

Proposition: Minkowski's Inequality

Let $p\in[1,\infty)$. Let $x=(x_1,x_2,\ldots x_n)$ and $y=(y_1,y_2,\ldots y_n)$ be two vectors of a vector space \(V\) over the field of real numbers \(\mathbb R\) or the field of complex numbers \(\mathbb C\). Then the sum of the p-norms $||x||_p+||y||_p$ can be used as an upper bound of the p-norm of the vector sum $||x+y||_p$, formally:

$$||x+y||_p\le ||x||_p+||y||_p.$$

Proofs: 1

Proofs: 1

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  1. Forster Otto: "Analysis 1, Differential- und Integralrechnung einer Veränderlichen", Vieweg Studium, 1983