Minkowski inequality

In mathematical analysis, the Minkowski inequality establishes that the L^p spaces satisfy the triangle inequality in the definition of normed vector spaces. The inequality is named after the German mathematician Hermann Minkowski.

Let ${\textstyle S}$ be a measure space, let ${\textstyle 1\leq p\leq \infty }$ and let ${\textstyle f}$ and ${\textstyle g}$ be elements of ${\textstyle L^{p}(S).}$ Then ${\textstyle f+g}$ is in ${\textstyle L^{p}(S),}$ and we have the triangle inequality

$${\displaystyle \|f+g\|_{p}\leq \|f\|_{p}+\|g\|_{p}}$$

with equality for ${\textstyle 1<p<\infty }$ if and only if ${\textstyle f}$ and ${\textstyle g}$ are positively linearly dependent; that is, ${\textstyle f=\lambda g}$ for some ${\textstyle \lambda \geq 0}$ or ${\textstyle g=0.}$ Here, the norm is given by:

$${\displaystyle \|f\|_{p}=\left(\int |f|^{p}d\mu \right)^{\frac {1}{p}}}$$

if ${\textstyle p<\infty ,}$ or in the case ${\textstyle p=\infty }$ by the essential supremum

$${\displaystyle \|f\|_{\infty }=\operatorname {ess\ sup} _{x\in S}|f(x)|.}$$

The Minkowski inequality is the triangle inequality in ${\textstyle L^{p}(S).}$ In fact, it is a special case of the more general fact

$${\displaystyle \|f\|_{p}=\sup _{\|g\|_{q}=1}\int |fg|d\mu ,\qquad {\tfrac {1}{p}}+{\tfrac {1}{q}}=1}$$

where it is easy to see that the right-hand side satisfies the triangular inequality.

Like Hölder's inequality, the Minkowski inequality can be specialized to sequences and vectors by using the counting measure:

$${\displaystyle {\biggl (}\sum _{k=1}^{n}|x_{k}+y_{k}|^{p}{\biggr )}^{1/p}\leq {\biggl (}\sum _{k=1}^{n}|x_{k}|^{p}{\biggr )}^{1/p}+{\biggl (}\sum _{k=1}^{n}|y_{k}|^{p}{\biggr )}^{1/p}}$$

for all real (or complex) numbers ${\textstyle x_{1},\dots ,x_{n},y_{1},\dots ,y_{n}}$ and where ${\textstyle n}$ is the cardinality of ${\textstyle S}$ (the number of elements in ${\textstyle S}$).

In probabilistic terms, given the probability space ${\displaystyle (\Omega ,{\mathcal {F}},\mathbb {P} ),}$ and ${\displaystyle \mathbb {E} }$ denote the expectation operator for every real- or complex-valued random variables ${\displaystyle X}$ and ${\displaystyle Y}$ on ${\displaystyle \Omega ,}$ Minkowski's inequality reads

${\displaystyle \left(\mathbb {E} [|X+Y|^{p}]\right)^{\frac {1}{p}}\leqslant \left(\mathbb {E} [|X|^{p}]\right)^{\frac {1}{p}}+\left(\mathbb {E} [|Y|^{p}]\right)^{\frac {1}{p}}.}$