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In mathematical analysis, the Minkowski inequality establishes that the Lp spaces are normed vector spaces. Let S be a measure space, let 1 ≤ p ≤ ∞ and let f and g be elements of Lp(S). Then f + g is in Lp(S), and we have the triangle inequality

\( \|f+g\|_p \le \|f\|_p + \|g\|_p \)

with equality for 1 < p < ∞ if and only if f and g are positively linearly dependent, i.e., \( f = \lambda g \) for some \( \lambda \) ≥ 0. Here, the norm is given by:

\( \|f\|_p = \left( \int |f|^p d\mu \right)^{1/p} \)

if p < ∞, or in the case p = ∞ by the essential supremum

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

The Minkowski inequality is the triangle inequality in Lp(S). In fact, it is a special case of the more general fact

\( \|f\|_p = \sup_{\|g\|_q = 1} \int |fg| d\mu, \qquad 1/p + 1/q = 1 \)

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:

\( \left( \sum_{k=1}^n |x_k + y_k|^p \right)^{1/p} \le \left( \sum_{k=1}^n |x_k|^p \right)^{1/p} + \left( \sum_{k=1}^n |y_k|^p \right)^{1/p} \)

for all real (or complex) numbers \( x_1, ..., x_n, y_1, ..., y_n \) and where n is the cardinality of S (the number of elements in S).

Proof

First, we prove that f+g has finite p-norm if f and g both do, which follows by

\( |f + g|^p \le 2^{p-1}(|f|^p + |g|^p). \)

Indeed, here we use the fact that \( h(x)=x^p is convex over \( \mathbb{R}^+ (for p greater than one) and so, if a and b are both positive then, by Jensen's inequality,

\( \left(\frac{1}{2} a + \frac{1}{2} b\right)^p \le \frac{1}{2}a^p + \frac{1}{2} b^p. \)

This means that

\( (a+b)^p \le 2^{p-1}a^p + 2^{p-1}b^p. \)

Now, we can legitimately talk about \( (\|f + g\|_p) \) . If it is zero, then Minkowski's inequality holds. We now assume that (\|f + g\|_p) \) is not zero. Using Hölder's inequality

\( \|f + g\|_p^p = \int |f + g|^p \, \mathrm{d}\mu \)
\( \le \int (|f| + |g|)|f + g|^{p-1} \, \mathrm{d}\mu \)
\( =\int |f||f + g|^{p-1} \, \mathrm{d}\mu+\int |g||f + g|^{p-1} \, \mathrm{d}\mu \)
\( \stackrel{\text{H}\ddot{\text{o}}\text{lder}}{\le} \left( \left(\int |f|^p \, \( \mathrm{d}\mu\right)^{1/p} + \left (\int |g|^p \,\mathrm{d}\mu\right)^{1/p} \right) \left(\int |f + g|^{(p-1)\left(\frac{p}{p-1}\right)} \, \mathrm{d}\mu \right)^{1-\frac{1}{p}} \)
\( = (\|f\|_p + \|g\|_p)\frac{\|f + g\|_p^p}{\|f + g\|_p}. \)

We obtain Minkowski's inequality by multiplying both sides by \( \frac{\|f + g\|_p}{\|f + g\|_p^p}. \)
Minkowski's integral inequality

Suppose that (S1,μ1) and (S2,μ2) are two measure spaces and F : S1×S2 → R is measurable. Then Minkowski's integral inequality is (Stein 1970, §A.1), (Hardy, Littlewood & Pólya 1988, Theorem 202):

\( \left[\int_{S_2}\left|\int_{S_1}F(x,y)\,d\mu_1(x)\right|^pd\mu_2(y)\right]^{1/p} \le \int_{S_1}\left(\int_{S_2}|F(x,y)|^p\,d\mu_2(y)\right)^{1/p}d\mu_1(x), \)

with obvious modifications in the case p = ∞. If p > 1, and both sides are finite, then equality holds only if |F(x,y)| = φ(x)ψ(y) a.e. for some non-negative measurable functions φ and ψ.

If μ1 is the counting measure on a two-point set S1 = {1,2}, then Minkowski's integral inequality gives the usual Minkowski inequality as a special case: for putting ƒi(y) = F(i,y) for i = 1,2, the integral inequality gives

\( \begin{align} \|f_1 + f_2\|_p &= \left[\int_{S_2}\left|\int_{S_1}F(x,y)\,d\mu_1(x)\right|^pd\mu_2(y)\right]^{1/p} \\ &\le\int_{S_1}\left(\int_{S_2}|F(x,y)|^p\,d\mu_2(y)\right)^{1/p}d\mu_1(x)\\ &=\|f_1\|_p + \|f_2\|_p. \end{align} \)

See also

Mahler's inequality
Hölder's inequality

References

Hardy, G. H.; Littlewood, J. E.; Pólya, G. (1952). Inequalities. Cambridge Mathematical Library (second ed.). Cambridge: Cambridge University Press. ISBN 0-521-35880-9.
Minkowski, H. (1953). Geometrie der Zahlen. Chelsea.
Stein, Elias (1970). Singular integrals and differentiability properties of functions. Princeton University Press.
M.I. Voitsekhovskii (2001), "Minkowski inequality", in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer, ISBN 978-1556080104
Arthur Lohwater (1982). "Introduction to Inequalities". Online e-book in PDF format.

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