View source for Minkowski inequality

{{Short description|Inequality that established ''L''<sup>''p''</sup> spaces are normed vector spaces}}

In [[Mathematical analysis|mathematical analysis]], the '''Minkowski inequality''' establishes that the [[Lp space|L<sup>''p''</sup> spaces]] are [[Normed vector space|normed vector space]]s. Let <math>S</math> be a [[Measure space|measure space]], let <math>1 \leq p < \infty</math> and let <math>f</math> and <math>g</math> be elements of <math>L^p(S).</math> Then <math>f + g</math> is in <math>L^p(S),</math> and we have the [[Triangle inequality|triangle inequality]]
<math display=block>\|f+g\|_p \leq \|f\|_p + \|g\|_p</math>
with equality for <math>1 < p < \infty</math> if and only if <math>f</math> and <math>g</math> are positively linearly dependent; that is, <math>f = \lambda g</math> for some <math>\lambda \geq 0</math> or <math>g = 0.</math> Here, the norm is given by:
<math display=block>\|f\|_p = \left(\int |f|^p d\mu\right)^{\frac{1}{p}}</math>
if <math>p < \infty,</math> or in the case <math>p = \infty</math> by the essential supremum
<math display=block>\|f\|_\infty = \operatorname{ess\ sup}_{x\in S}|f(x)|.</math>

The Minkowski inequality is the triangle inequality in <math>L^p(S).</math> In fact, it is a special case of the more general fact
<math display=block>\|f\|_p = \sup_{\|g\|_q = 1} \int |fg| d\mu, \qquad \tfrac{1}{p} + \tfrac{1}{q} = 1</math>
<!--, which is probably easier to prove -->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|counting measure]]:
<math display=block>\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}</math>
for all [[Real number|real]] (or [[Complex number|complex]]) numbers <math>x_1, \dots, x_n, y_1, \dots, y_n</math> and where <math>n</math> is the [[Cardinality|cardinality]] of <math>S</math> (the number of elements in <math>S</math>).

The inequality is named after the German mathematician [[Biography:Hermann Minkowski|Hermann Minkowski]].

==Proof==

First, we prove that <math>f + g</math> has finite <math>p</math>-norm if <math>f</math> and <math>g</math> both do, which follows by
<math display=block>|f + g|^p \leq 2^{p-1}(|f|^p + |g|^p).</math>
Indeed, here we use the fact that <math>h(x) = |x|^p</math> is [[Convex function|convex]] over <math>\Reals^+</math> (for <math>p > 1</math>) and so, by the definition of convexity,
<math display=block>\left|\tfrac{1}{2} f + \tfrac{1}{2} g\right|^p \leq \left|\tfrac{1}{2} |f| + \tfrac{1}{2} |g|\right|^p \leq \tfrac{1}{2}|f|^p + \tfrac{1}{2} |g|^p.</math>
This means that
<math display=block>|f+g|^p \leq \tfrac{1}{2}|2f|^p + \tfrac{1}{2}|2g|^p = 2^{p-1}|f|^p + 2^{p-1}|g|^p.</math>

Now, we can legitimately talk about <math>\|f + g\|_p.</math> If it is zero, then Minkowski's inequality holds. We now assume that <math>\|f + g\|_p</math> is not zero. Using the triangle inequality and then [[Hölder's inequality]], we find that
<math display=block>\begin{align}
\|f + g\|_p^p &= \int |f + g|^p \, \mathrm{d}\mu \\
&= \int |f + g| \cdot |f + g|^{p-1} \, \mathrm{d}\mu \\
&\leq \int (|f| + |g|)|f + g|^{p-1} \, \mathrm{d}\mu \\
&=\int |f||f + g|^{p-1} \, \mathrm{d}\mu+\int |g||f + g|^{p-1} \, \mathrm{d}\mu \\
&\leq \left(\left(\int |f|^p \, \mathrm{d}\mu\right)^{\frac{1}{p}} + \left(\int |g|^p \,\mathrm{d}\mu\right)^{\frac{1}{p}}\right)\left(\int |f + g|^{(p-1)\left(\frac{p}{p-1}\right)} \, \mathrm{d}\mu\right)^{1-\frac{1}{p}} && \text{ Hölder's inequality} \\
&= \left(\|f\|_p + \|g\|_p \right )\frac{\|f + g\|_p^p}{\|f + g\|_p}
\end{align}</math>

We obtain Minkowski's inequality by multiplying both sides by
<math display=block>\frac{\|f + g\|_p}{\|f + g\|_p^p}.</math>

==Minkowski's integral inequality==

Suppose that <math>(S_1, \mu_1)</math> and <math>(S_2, \mu_2)</math> are two {{sigma}}-finite measure spaces and <math>F : S_1 \times S_2 \to \Reals</math> is measurable.  Then Minkowski's integral inequality is {{harv|Stein|1970|loc=§A.1}}, {{harv|Hardy|Littlewood|Pólya|1988|loc=Theorem 202}}:
<math display=block>\left[\int_{S_2}\left|\int_{S_1}F(x,y)\, \mu_1(\mathrm{d}x)\right|^p \mu_2(\mathrm{d}y)\right]^{\frac{1}{p}} ~\leq~ \int_{S_1}\left(\int_{S_2}|F(x,y)|^p\,\mu_2(\mathrm{d}y)\right)^{\frac{1}{p}}\mu_1(\mathrm{d}x),</math>
with obvious modifications in the case <math>p = \infty.</math>  If <math>p > 1,</math> and both sides are finite, then equality holds only if <math>|F(x, y)| = \varphi(x) \, \psi(y)</math> a.e. for some non-negative measurable functions <math>\varphi</math> and <math>\psi.</math>

If <math>\mu_1</math> is the counting measure on a two-point set <math>S_1 = \{1, 2\},</math> then Minkowski's integral inequality gives the usual Minkowski inequality as a special case: for putting <math>f_i(y) = F(i, y)</math> for <math>i = 1, 2,</math> the integral inequality gives
<math display=block>\|f_1 + f_2\|_p = \left(\int_{S_2}\left|\int_{S_1}F(x,y)\,\mu_1(\mathrm{d}x)\right|^p \mu_2(\mathrm{d}y)\right)^{\frac{1}{p}} \leq \int_{S_1}\left(\int_{S_2}|F(x,y)|^p\,\mu_2(\mathrm{d}y)\right)^{\frac{1}{p}} \mu_1(\mathrm{d}x) = \|f_1\|_p + \|f_2\|_p.</math>

If the measurable function <math>F : S_1 \times S_2 \to \Reals</math> is non-negative then for all <math>1 \leq p \leq q \leq \infty,</math>{{sfn|Bahouri|Chemin|Danchin|2011|p=4}}
<math display=block>\left\|\left\|F(\,\cdot, s_2)\right\|_{L^p(S_1, \mu_1)}\right\|_{L^q(S_2, \mu_2)}
~\leq~ \left\|\left\|F(s_1, \cdot)\right\|_{L^q(S_2, \mu_2)}\right\|_{L^p(S_1, \mu_1)} \ .</math>

This notation has been generalized to
<math display=block>\|f\|_{p,q} = \left(\int_{\R^m} \left[\int_{\R^n}|f(x,y)|^q\mathrm{d}y\right]^{\frac{p}{q}} \mathrm{d}x\right)^{\frac{1}{p}}</math>
for <math>f : \R^{m+n} \to E,</math> with <math>\mathcal{L}_{p,q}(\R^{m+n},E) = \{f \in E^{\R^{m+n}} : \|f\|_{p,q} < \infty\}.</math>  Using this notation, manipulation of the exponents reveals that, if <math>p < q,</math> then <math>\|f\|_{q,p} \leq \|f\|_{p,q}.</math>

==Reverse inequality==

When <math>p < 1</math> the reverse inequality holds:
<math display=block>\|f+g\|_p \ge \|f\|_p + \|g\|_p.</math>

We further need the restriction that both <math>f</math> and <math>g</math> are non-negative, as we can see from the example <math>f=-1, g=1</math> and <math>p = 1:</math> <math>\|f+g\|_1 = 0 < 2 = \|f\|_1 + \|g\|_1.</math>

The reverse inequality follows from the same argument as the standard Minkowski, but uses that Holder's inequality is also reversed in this range.

Using the Reverse Minkowski, we may prove that power means with <math>p \leq 1,</math> such as the [[Harmonic mean|harmonic mean]] and the [[Geometric mean|geometric mean]] are concave.

==Generalizations to other functions==

The Minkowski inequality can be generalized to other functions <math>\phi(x)</math> beyond the power function
<math>x^p.</math> The generalized inequality has the form 
<math display=block>\phi^{-1}\left(\textstyle\sum\limits_{i=1}^n \phi(x_i + y_i)\right) \leq \phi^{-1}\left(\textstyle\sum\limits_{i=1}^n \phi(x_i)\right) + \phi^{-1}\left(\textstyle\sum\limits_{i=1}^n \phi(y_i)\right).</math>

Various sufficient conditions on <math>\phi</math> have been found by Mulholland<ref>{{cite journal|last=Mulholland|first=H.P.|title=On Generalizations of Minkowski's Inequality in the Form of a Triangle Inequality|journal=Proceedings of the London Mathematical Society|volume=s2-51|number=1|pages=294–307|year=1949|doi=10.1112/plms/s2-51.4.294}}</ref> and others. For example, for <math>x \geq 0</math> one set of sufficient conditions from Mulholland is 
# <math>\phi(x)</math> is continuous and strictly increasing with <math>\phi(0) = 0.</math>
# <math>\phi(x)</math> is a convex function of <math>x.</math>
# <math>\log\phi(x)</math> is a convex function of <math>\log(x).</math>

==See also==

* {{annotated link|Cauchy–Schwarz inequality}}
* {{annotated link|Hölder's inequality}}
* {{annotated link|Mahler's inequality}}
* {{annotated link|Young's convolution inequality}}
* {{annotated link|Young's inequality for products}}

==References==

{{reflist}}

* {{Bahouri Chemin Danchin Fourier Analysis and Nonlinear Partial Differential Equations 2011}} <!--{{sfn|Bahouri|Chemin|Danchin|2011|p=}}-->
* {{Cite book|last1=Hardy|first1=G. H.|last2=Littlewood|first2=J. E.|last3=Pólya|first3=G.|title=Inequalities|year=1988|url=https://archive.org/details/inequalities0000hard|url-access=registration|series=Cambridge Mathematical Library|edition=second|publisher=Cambridge University Press|location=Cambridge|isbn=0-521-35880-9}}
* {{Cite journal|last=Minkowski|first=H.|title=Geometrie der Zahlen|publisher=Chelsea|year=1953}}.
* {{Cite journal|last=Stein|first=Elias|title=Singular integrals and differentiability properties of functions|publisher=Princeton University Press|year=1970}}.
* {{springer|id=M/m064060|title=Minkowski inequality|author=M.I. Voitsekhovskii}}
* {{cite web|last=Lohwater|first=Arthur J.|title=Introduction to Inequalities|year=1982|url=http://www.mediafire.com/file/1mw1tkgozzu}}

==Further reading==

* {{Citation|last=Bullen|first=P. S.|title=The Power Means|date=2003|url=http://link.springer.com/10.1007/978-94-017-0399-4_3|work=Handbook of Means and Their Inequalities |pages=175–265|place=Dordrecht|publisher=Springer Netherlands|language=en|doi=10.1007/978-94-017-0399-4_3|isbn=978-90-481-6383-0|access-date=2022-06-23}}


[[Category:Articles containing proofs]]
[[Category:Inequalities]]
[[Category:Measure theory]]

{{Sourceattribution|Minkowski inequality}}