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 σ-finite 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)\,\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

\|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}.

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) [1994], "Minkowski inequality", in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4
Arthur Lohwater (1982). "Introduction to Inequalities". Missing or empty |url= (help)

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Minkowski inequality

Proof

Minkowski's integral inequality

See also

References