Minkowski inequality

This page is about Minkowski's inequality for norms. See Minkowski's first inequality for convex bodies for Minkowski's inequality in convex geometry.

In mathematical analysis, the Minkowski inequality establishes that the L^p spaces are normed vector spaces. Let S be a measure space, let 1 ≤ p ≤ ∞ and let f and g be elements of L^p(S). Then f + g is in L^p(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 = λg for some λ ≥ 0 or g = 0. Here, the norm is given by:

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

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 L^p(S). In fact, it is a special case of the more general fact

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

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:

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

for all real (or complex) numbers x₁, ..., x_n, y₁, ..., 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 R⁺ (for p > 1) and so, by the definition of convexity,

$\left|\tfrac{1}{2} f + \tfrac{1}{2} g\right|^p\le\left|\tfrac{1}{2} |f| + \tfrac{1}{2} |g|\right|^p \le \tfrac{1}{2}|f|^p + \tfrac{1}{2} |g|^p.$

This means that

$|f+g|^p \le \tfrac{1}{2}|2f|^p + \tfrac{1}{2}|2g|^p=2^{p-1}|f|^p + 2^{p-1}|g|^p.$

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 the triangle inequality and then Hölder's inequality, we find that

$\begin{align} \|f + g\|_p^p &= \int |f + g|^p \, \mathrm{d}\mu \\ &= \int |f + g| \cdot |f + g|^{p-1} \, \mathrm{d}\mu \\ &\le \int (|f| + |g|)|f + g|^{p-1} \, \mathrm{d}\mu \\ &=\int |f||f + g|^{p-1} \, \mathrm{d}\mu+\int |g||f + g|^{p-1} \, \mathrm{d}\mu \\ &\le \left( \left(\int |f|^p \, \mathrm{d}\mu\right)^{\frac{1}{p}} + \left (\int |g|^p \,\mathrm{d}\mu\right)^{\frac{1}{p}} \right) \left(\int |f + g|^{(p-1)\left(\frac{p}{p-1}\right)} \, \mathrm{d}\mu \right)^{1-\frac{1}{p}} && \text{ Hölder's inequality} \\ &= \left (\|f\|_p + \|g\|_p \right )\frac{\|f + g\|_p^p}{\|f + g\|_p} \end{align}$

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):

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

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 f_i(y) = F(i,y) for i = 1, 2, the integral inequality gives

$\|f_1 + f_2\|_p = \left(\int_{S_2}\left|\int_{S_1}F(x,y)\,d\mu_1(x)\right|^pd\mu_2(y)\right)^{\frac{1}{p}} \le\int_{S_1}\left(\int_{S_2}|F(x,y)|^p\,d\mu_2(y)\right)^{\frac{1}{p}}d\mu_1(x)=\|f_1\|_p + \|f_2\|_p.$

See also

• Mahler's inequality
• Hölder's inequality

References

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