Minkowski inequality

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:

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

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,

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

This means that

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

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

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

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):

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

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

${\displaystyle \|f_{1}+f_{2}\|_{p}=\left(\int _{S_{2}}\left|\int _{S_{1}}F(x,y)\,\mu _{1}(\mathrm {d} x)\right|^{p}\mu _{2}(\mathrm {d} y)\right)^{\frac {1}{p}}\leq \int _{S_{1}}\left(\int _{S_{2}}|F(x,y)|^{p}\,\mu _{2}(\mathrm {d} y)\right)^{\frac {1}{p}}\mu _{1}(\mathrm {d} x)=\|f_{1}\|_{p}+\|f_{2}\|_{p}.}$

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 978-1-55608-010-4
• Arthur Lohwater (1982). "Introduction to Inequalities".