Minkowski inequality

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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[edit]

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[edit]

Suppose that and are two 𝜎-finite measure spaces and is measurable. Then Minkowski's integral inequality is (Stein 1970, §A.1), (Hardy, Littlewood & Pólya 1988, Theorem 202):

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 [1]

This notation has been generalized to

for with Using this notation, manipulation of the exponents reveals that, if then

Reverse inequality[edit]

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[edit]

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[2] 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[edit]

References[edit]

  1. ^ Bahouri, Chemin & Danchin 2011, p. 4.
  2. ^ 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[edit]