In mathematical analysis, the Minkowski inequality establishes that the Lp spaces are normed vector spaces. Let be a measure space, let and let and be elements of Then is in and we have the triangle inequality with equality for if and only if and are positively linearly dependent; that is, for some or Here, the norm is given by: if or in the case by the essential supremum

The Minkowski inequality is the triangle inequality in 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 and where is the cardinality of (the number of elements in ).

The inequality is named after the German mathematician Hermann Minkowski.

Proof

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First, we prove that   has finite  -norm if   and   both do, which follows by   Indeed, here we use the fact that   is convex over   (for  ) 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

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Suppose that   and   are two 𝜎-finite measure spaces and   is measurable. Then Minkowski's integral inequality is:[1][2]   with obvious modifications in the case   If   and both sides are finite, then equality holds only if   a.e. for some non-negative measurable functions   and  

If   is the counting measure on a two-point set   then Minkowski's integral inequality gives the usual Minkowski inequality as a special case: for putting   for   the integral inequality gives  

If the measurable function   is non-negative then for all  [3]  

This notation has been generalized to   for   with   Using this notation, manipulation of the exponents reveals that, if   then  

Reverse inequality

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When   the reverse inequality holds:  

We further need the restriction that both   and   are non-negative, as we can see from the example   and    

The reverse inequality follows from the same argument as the standard Minkowski, but uses that Holder's inequality is also reversed in this range.

Using the Reverse Minkowski, we may prove that power means with   such as the harmonic mean and the geometric mean are concave.

Generalizations to other functions

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The Minkowski inequality can be generalized to other functions   beyond the power function   The generalized inequality has the form  

Various sufficient conditions on   have been found by Mulholland[4] and others. For example, for   one set of sufficient conditions from Mulholland is

  1.   is continuous and strictly increasing with  
  2.   is a convex function of  
  3.   is a convex function of  

See also

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References

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  1. ^ Stein 1970, §A.1.
  2. ^ Hardy, Littlewood & Pólya 1988, Theorem 202.
  3. ^ Bahouri, Chemin & Danchin 2011, p. 4.
  4. ^ Mulholland, H.P. (1949). "On Generalizations of Minkowski's Inequality in the Form of a Triangle Inequality". Proceedings of the London Mathematical Society. s2-51 (1): 294–307. doi:10.1112/plms/s2-51.4.294.

Further reading

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