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 <math display="inline">S</math> be a measure space, let <math display="inline">1 \leq p \leq \infty</math> and let <math display="inline">f</math> and <math display="inline">g</math> be elements of <math display="inline">L^p(S).</math> Then <math display="inline">f + g</math> is in <math display="inline">L^p(S),</math> and we have the triangle inequality

<math display="block">\|f+g\|_p \leq \|f\|_p + \|g\|_p</math>

with equality for <math display="inline">1 < p < \infty</math> if and only if <math display="inline">f</math> and <math display="inline">g</math> are positively linearly dependent; that is, <math display="inline">f = \lambda g</math> for some <math display="inline">\lambda \geq 0</math> or <math display="inline">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 display="inline">p < \infty,</math> or in the case <math display="inline">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 display="inline">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>

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:

<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 (or complex) numbers <math display="inline">x_1, \dots, x_n, y_1, \dots, y_n</math> and where <math display="inline">n</math> is the cardinality of <math display="inline">S</math> (the number of elements in <math display="inline">S</math>).

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

<math>\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}}.</math>

Proof

Proof by Hölder's inequality

First, we prove that <math display="inline">f + g</math> has finite <math display="inline">p</math>-norm if <math display="inline">f</math> and <math display="inline">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 display="inline">h(x) = |x|^p</math> is convex over <math display="inline">\Reals^+</math> (for <math display="inline">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 display="inline">\|f + g\|_p.</math> If it is zero, then Minkowski's inequality holds. We now assume that <math display="inline">\|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>

Proof by a direct convexity argument

Given <math>t \in (0, 1)</math>, one has, by convexity (Jensen's inequality), for every <math>x \in S</math>

<math>

 |f (x) + g (x)|^p = \Bigl|(1-t) \frac{f(x)}{1 -t} + t \frac{g(x)}{t} \Bigr|^p

\le (1-t) \Bigl|\frac{f(x)}{1 -t}\Bigr|^p + t \Bigl|\frac{g(x)}{t} \Bigr|^p = \frac{|f(x)|^p}{(1-t)^{p - 1}} + \frac{|g(x)|^p}{t^{p-1}}. </math> By integration this leads to

<math>

 \int_{S} |f + g|^p\, \mathrm{d}\mu

\le \frac{1}{(1-t)^{p-1}} \int_{S} |f|^p\, \mathrm{d}\mu + \frac{1}{t^{p-1}} \int_{S} |g|^p\, \mathrm{d}\mu. </math> One takes then

<math>

t = \frac{\Vert g \Vert_{p}}{\Vert f \Vert_{p} + \Vert g \Vert_{p}}

</math> to reach the conclusion.

Minkowski's integral inequality

Suppose that <math display="inline">(S_1, \mu_1)</math> and <math display="inline">(S_2, \mu_2)</math> are two Template:Sigma-finite measure spaces and <math display="inline">F : S_1 \times S_2 \to \Reals</math> is measurable. Then Minkowski's integral inequality is:Template:SfnTemplate:Sfn

<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),\quad p\in[1,\infty)</math>

with obvious modifications in the case <math display="inline">p = \infty.</math> If <math display="inline">p > 1,</math> and both sides are finite, then equality holds only if <math display="inline">|F(x, y)| = \varphi(x) \, \psi(y)</math> a.e. for some non-negative measurable functions <math display="inline">\varphi</math> and <math display="inline">\psi.</math>

If <math display="inline">\mu_1</math> is the counting measure on a two-point set <math display="inline">S_1 = \{1, 2\},</math> then Minkowski's integral inequality gives the usual Minkowski inequality as a special case: for putting <math display="inline">f_i(y) = F(i, y)</math> for <math display="inline">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 display="inline">F : S_1 \times S_2 \to \Reals</math> is non-negative then for all <math display="inline">1 \leq p \leq q \leq \infty,</math>Template:Sfn

<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 display="inline">f : \R^{m+n} \to E,</math> with <math display="inline">\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 display="inline">p < q,</math> then <math display="inline">\|f\|_{q,p} \leq \|f\|_{p,q}.</math>

Reverse inequality

When <math display="inline">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 display="inline">f</math> and <math display="inline">g</math> are non-negative, as we can see from the example <math display="inline">f=-1, g=1</math> and <math display="inline">p = 1:</math> <math display="inline">\|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 display="inline">p \leq 1,</math> such as the harmonic mean and the geometric mean are concave.

Generalizations to other functions

The Minkowski inequality can be generalized to other functions <math display="inline">\phi(x)</math> beyond the power function <math display="inline">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 display="inline">\phi</math> have been found by Mulholland<ref>Template:Cite journal</ref> and others. For example, for <math display="inline">x \geq 0</math> one set of sufficient conditions from Mulholland is

1. <math display="inline">\phi(x)</math> is continuous and strictly increasing with <math display="inline">\phi(0) = 0.</math>
2. <math display="inline">\phi(x)</math> is a convex function of <math display="inline">x.</math>
3. <math display="inline">\log\phi(x)</math> is a convex function of <math display="inline">\log(x).</math>

See also

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References

• Template:Bahouri Chemin Danchin Fourier Analysis and Nonlinear Partial Differential Equations 2011
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Further reading

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