# 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

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:

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

The Minkowski inequality is the triangle inequality in L^{p}(*S*). In fact, it is a special case of the more general fact

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:

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

Indeed, here we use the fact that is convex over **R**^{+} (for *p* > 1) and so, by the definition of convexity,

This means that

Now, we can legitimately talk about . If it is zero, then Minkowski's inequality holds. We now assume that is not zero. Using the triangle inequality and then Hölder's inequality, we find that

We obtain Minkowski's inequality by multiplying both sides by

## Minkowski's integral inequality

Suppose that (*S*_{1}, *μ*_{1}) and (*S*_{2}, *μ*_{2}) are two *σ*-finite measure spaces and *F*′ : *S*_{1} × *S*_{2} → **R** is measurable. Then Minkowski's integral inequality is (Stein 1970, §A.1), (Hardy, Littlewood & Pólya 1988, Theorem 202):

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 μ_{1} is the counting measure on a two-point set *S*_{1} = {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

This notation has been generalized to

for , with .

## 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 Cite journal requires
`|journal=`

(help). - Stein, Elias (1970). "Singular integrals and differentiability properties of functions". Princeton University Press Cite journal requires
`|journal=`

(help). - M.I. Voitsekhovskii (2001) [1994], "Minkowski inequality", in Hazewinkel, Michiel (ed.),
*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)