# Minkowski's Inequality for Integrals

## Theorem

Let $f, g$ be integrable functions in $X \subseteq \R^n$ with respect to the volume element $dV$.

$(1):\quad$ Let $p > 1$. Then:
$\displaystyle \paren {\int_X \size {f + g}^p \rd V}^{1/p} \le \paren {\int_X \size f^p \rd V}^{1/p} + \paren {\int_X \size g^p \rd V}^{1/p}$
$(2):\quad$ Let $p < 1, p \ne 0$. Then:
$\displaystyle \paren {\int_X \size {f + g}^p \rd V}^{1/p} \ge \paren {\int_X \size f^p \rd V}^{1/p} + \paren {\int_X \size g^p \rd V}^{1/p}$

## Source of Name

This entry was named for Hermann Minkowski.