Minkowski inequality

The Minkowski inequality , also known as Minkowski inequality or inequality of Minkowski called, is an inequality in the border area between the measure theory and functional analysis , two sub-areas of mathematics . It is formulated in different versions, mostly for the sequence space as well as the Lebesgue spaces and . In these spaces it corresponds to the triangle inequality and thus makes them normalized rooms (in the case of a semi-normalized room ). ${\displaystyle \ell ^{p}}$ ${\displaystyle L^{p}}$ ${\displaystyle {\mathcal {L}}^{p}}$ ${\displaystyle {\mathcal {L}}^{p}}$

It is named after Hermann Minkowski , who first showed the inequality for infinite sums in 1896 in the first volume of his Geometry of Numbers .

Formulation for L ^p spaces

Let and the corresponding L ^p -space . It is the corresponding norm. So for one is ${\displaystyle p\in [1,\infty ]}$ ${\displaystyle L^{p}=L^{p}(\Omega ,{\mathcal {A}},\mu )}$ ${\displaystyle \|\cdot \|_{L^{p}}}$ ${\displaystyle L^{p}}$ ${\displaystyle f\in L^{p}}$

${\displaystyle \|f\|_{L^{p}}:={\begin{cases}\left(\int _{\Omega }|f|^{p}\mathrm {d} \mu \right)^{\tfrac {1}{p}}&{\text{ für }}p\in [1,\infty )\\\mathrm {ess} \sup _{x\in \Omega }|f(x)|&{\text{ für }}p=\infty \end{cases}}}$.

Here denotes the essential supremum . The Minkowski inequality then says: ${\displaystyle \mathrm {ess} \sup }$

Is and so is true ${\displaystyle f\in L^{p}}$ ${\displaystyle g\in L^{p}}$

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

The inequality also applies in (see Lp space # definition ). The (semi) norm is defined identically to the norm, but denoted by. The Minkowski inequality then says: ${\displaystyle {\mathcal {L}}^{p}={\mathcal {L}}^{p}(\Omega ,{\mathcal {A}},\mu )}$ ${\displaystyle {\mathcal {L}}^{p}}$ ${\displaystyle L^{p}}$ ${\displaystyle \|\cdot \|_{{\mathcal {L}}^{p}}}$

Is and so is true ${\displaystyle f\in {\mathcal {L}}^{p}}$ ${\displaystyle g\in {\mathcal {L}}^{p}}$

${\displaystyle \|f+g\|_{{\mathcal {L}}^{p}}\leq \|f\|_{{\mathcal {L}}^{p}}+\|g\|_{{\mathcal {L}}^{p}}}$.

Formulation for measurable functions

The Minkowski inequality can also be formulated more generally for measurable functions . The agreements for define ${\displaystyle \infty ^{p}=\infty ,\;\infty ^{-p}=0}$ ${\displaystyle p\in (0,\infty )}$

${\displaystyle I_{p}(f):={\begin{cases}\left(\int _{X}|f|^{p}\mathrm {d} \mu \right)^{\tfrac {1}{p}}&{\text{ für }}p\in [1,\infty )\\\mathrm {ess} \sup _{x\in X}|f(x)|&{\text{ für }}p=\infty \end{cases}}}$,

where is a measurable function of the measure space after . Here is or . Then the Minkwoski inequality reads: ${\displaystyle f}$ ${\displaystyle (X,{\mathcal {A}},\mu )}$ ${\displaystyle {\overline {\mathbb {K} }}}$ ${\displaystyle \mathbb {K} =\mathbb {R} }$ ${\displaystyle \mathbb {K} =\mathbb {C} }$

If the functions from to both are measurable, then applies ${\displaystyle f,g}$ ${\displaystyle X}$ ${\displaystyle {\overline {\mathbb {K} }}}$

${\displaystyle I_{p}(f+g)\leq I_{p}(f)+I_{p}(g)}$.

Formulation for consequences

The Minkowski inequality also holds for sequences in or in , regardless of whether the sequences converge. It then reads ${\displaystyle \mathbb {R} }$ ${\displaystyle \mathbb {C} }$

${\displaystyle \left(\sum _{k=1}^{\infty }|x_{k}+y_{k}|^{p}\right)^{1/p}\leq \left(\sum _{k=1}^{\infty }|x_{k}|^{p}\right)^{1/p}+\left(\sum _{k=1}^{\infty }|y_{k}|^{p}\right)^{1/p}}$

for . ${\displaystyle p\in [1,\infty )}$

If you limit yourself to the appropriate sequence space with the norm ${\displaystyle \ell ^{p}}$

${\displaystyle \|(x_{n})_{n}\|_{\ell ^{p}}:=\left(\sum _{n=1}^{\infty }|x_{n}|^{p}\right)^{\frac {1}{p}}}$,

this is the Minkowski inequality

${\displaystyle \|(x_{n})_{n}+(y_{n})_{n}\|_{\ell ^{p}}\leq \|(x_{n})_{n}\|_{\ell ^{p}}+\|(y_{n})_{n}\|_{\ell ^{p}}}$.

for the consequences of . This can be viewed as a special case of the inequality for the if one chooses the natural numbers as the basic set and the measure of count as the measure . ${\displaystyle (x_{n})_{n\in \mathbb {N} },(y_{n})_{n\in \mathbb {N} }}$ ${\displaystyle \ell ^{p}}$ ${\displaystyle L^{p}}$

proof

The Minkowski inequality is trivial for and . So be it . Since is a convex function , holds ${\displaystyle p=1}$ ${\displaystyle p=\infty }$ ${\displaystyle 1<p<\infty }$ ${\displaystyle x\mapsto |x|^{p}}$

${\displaystyle |f+g|^{p}=2^{p}\cdot \left|{\frac {1}{2}}\,f+{\frac {1}{2}}\,g\right|^{p}\leq 2^{p-1}(|f|^{p}+|g|^{p})}$

and therefore . ${\displaystyle f+g\in L^{p}(S)}$

Be in the following without loss of generality . The following applies: ${\displaystyle \|f+g\|_{p}>0}$

${\displaystyle {\begin{aligned}|f+g|^{p}&=(|f+g|)(|f+g|)^{p-1}\\&\leq (|f|+|g|)(|f+g|)^{p-1}\\&=\left(|f|\cdot |f+g|^{p-1}\right)+\left(|g|\cdot |f+g|^{p-1}\right)\\\end{aligned}}}$

Be . Then q is the Hölder exponent conjugated to p , the following applies: ${\displaystyle q:={\tfrac {p}{p-1}}}$ ${\displaystyle {\tfrac {1}{p}}+{\tfrac {1}{q}}=1}$

According to the Hölder inequality :

${\displaystyle {\begin{aligned}\|f+g\|_{p}^{p}=\int _{S}|f+g|^{p}&\leq \int _{S}\left(|f|\cdot |f+g|^{p-1}\right)+\int _{S}\left(|g|\cdot |f+g|^{p-1}\right)\\&\leq \|f\|_{p}\cdot \||f+g|^{p-1}\|_{q}+\|g\|_{p}\cdot \||f+g|^{p-1}\|_{q}\\[.2em]&=(\|f\|_{p}+\|g\|_{p})\cdot \||f+g|^{p-1}\|_{q}\\&=(\|f\|_{p}+\|g\|_{p})\cdot \left(\int _{S}|f+g|^{(p-1)\cdot {\frac {p}{p-1}}}\right)^{1-{\frac {1}{p}}}\\&=(\|f\|_{p}+\|g\|_{p})\cdot {\frac {\int _{S}|f+g|^{p}}{\left(\int _{S}|f+g|^{p}\right)^{\frac {1}{p}}}}\\&=(\|f\|_{p}+\|g\|_{p})\cdot {\frac {\|f+g\|_{p}^{p}}{\|f+g\|_{p}}},\\\end{aligned}}}$

This implies the Minkowski inequality after multiplying both sides by . ${\displaystyle {\tfrac {\|f+g\|_{p}}{\|f+g\|_{p}^{p}}}}$

Generalization (Minkowski inequality for integrals)

Let and be two measurement spaces and a measurable function, then (Minkowski inequality for integrals): ${\displaystyle (S_{1},\mu _{1})}$ ${\displaystyle (S_{2},\mu _{2})}$ ${\displaystyle F:S_{1}\times S_{2}\to \mathbb {K} }$

${\displaystyle \left[\int _{S_{2}}\left(\int _{S_{1}}|F(x,y)|\,d\mu _{1}(x)\right)^{p}d\mu _{2}(y)\right]^{1/p}\leq \int _{S_{1}}\left(\int _{S_{2}}|F(x,y)|^{p}\,d\mu _{2}(y)\right)^{1/p}d\mu _{1}(x),}$

for . If and both sides are finite, then equality applies precisely if and can be written as the product of two measurable functions . ${\displaystyle p<\infty }$ ${\displaystyle 1<p<\infty }$ ${\displaystyle |F|}$ ${\displaystyle |F|(x,y)=\varphi (x)\psi (y)}$ ${\displaystyle \phi \colon S_{1}\to [0,\infty )}$ ${\displaystyle \psi \colon S_{2}\to [0,\infty )}$

If we choose the two-element set with the counting measure, we get the usual Minkowski inequality again as a special case , namely with for is ${\displaystyle (S_{1},\mu _{1})}$ ${\displaystyle \{1,2\}}$ ${\displaystyle f_{i}=F(i,\,\cdot \,)}$ ${\displaystyle i=1,2}$

${\displaystyle {\begin{aligned}\|f_{1}+f_{2}\|_{p}&=\left[\int _{S_{2}}\left|\int _{S_{1}}F(x,y)\,d\mu _{1}(x)\right|^{p}d\mu _{2}(y)\right]^{1/p}\\&\leq \int _{S_{1}}\left(\int _{S_{2}}|F(x,y)|^{p}\,d\mu _{2}(y)\right)^{1/p}d\mu _{1}(x)\\&=\|f_{1}\|_{p}+\|f_{2}\|_{p}.\end{aligned}}}$

Web links

• MI Voitsekhovskii: Minkowski inequality . In: Michiel Hazewinkel (Ed.): Encyclopaedia of Mathematics . Springer-Verlag , Berlin 2002, ISBN 978-1-55608-010-4 (English, online ).
• Eric W. Weisstein : Minkowski's Inequalities . In: MathWorld (English).

literature

• Herbert Amann, Joachim Escher: Analysis III . 1st edition. Birkhäuser-Verlag Basel Boston Berlin, 2001, ISBN 3-7643-6613-3

Individual evidence

1. ↑ ^{a b c} Jürgen Elstrodt : Measure and integration theory . 6th, corrected edition. Springer-Verlag, Berlin Heidelberg 2009, ISBN 978-3-540-89727-9 , pp.  224-226 , doi : 10.1007 / 978-3-540-89728-6 .
2. ^ Hans Wilhelm Alt : Linear functional analysis . 6th edition. Springer-Verlag, Berlin Heidelberg 2012, ISBN 978-3-642-22260-3 , p.  57 , doi : 10.1007 / 978-3-642-22261-0 .
3. ↑ Achim Klenke: Probability Theory . 3. Edition. Springer-Verlag, Berlin Heidelberg 2013, ISBN 978-3-642-36017-6 , p. 154 , doi : 10.1007 / 978-3-642-36018-3 .