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Minkowski inequality

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In mathematical analysis, the Minkowski inequality establishes that the Lp spaces are normed vector spaces. Let S be a measure space, let 1 ≤ p ≤ ∞ and let f and g be elements of Lp(S). Then f + g is in Lp(S), and we have the triangle inequality

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∥ f + g ∥ p ≤ ∥ 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 = ( ∫ | f | p d μ ) 1 p

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

∥ f ∥ ∞ = e s s   s u p x ∈ S ⁡ | f ( x ) | .

The Minkowski inequality is the triangle inequality in Lp(S). In fact, it is a special case of the more general fact

∥ f ∥ p = sup ∥ g ∥ q = 1 ∫ | f g | d μ , 1 p + 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:

( ∑ k = 1 n | x k + y k | p ) 1 p ≤ ( ∑ k = 1 n | x k | p ) 1 p + ( ∑ k = 1 n | y k | p ) 1 p

for all real (or complex) numbers x1, ..., xn, y1, ..., yn 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 ≤ 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,

| 1 2 f + 1 2 g | p ≤ | 1 2 | f | + 1 2 | g | | p ≤ 1 2 | f | p + 1 2 | g | p .

This means that

| f + g | p ≤ 1 2 | 2 f | p + 1 2 | 2 g | 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

∥ f + g ∥ p p = ∫ | f + g | p d μ = ∫ | f + g | ⋅ | f + g | p − 1 d μ ≤ ∫ ( | f | + | g | ) | f + g | p − 1 d μ = ∫ | f | | f + g | p − 1 d μ + ∫ | g | | f + g | p − 1 d μ ≤ ( ( ∫ | f | p d μ ) 1 p + ( ∫ | g | p d μ ) 1 p ) ( ∫ | f + g | ( p − 1 ) ( p p − 1 ) d μ ) 1 − 1 p  Hölder's inequality = ( ∥ f ∥ p + ∥ g ∥ p ) ∥ f + g ∥ p p ∥ f + g ∥ p

We obtain Minkowski's inequality by multiplying both sides by

∥ 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):

[ ∫ S 2 | ∫ S 1 F ( x , y ) μ 1 ( d x ) | p μ 2 ( d y ) ] 1 p ≤ ∫ S 1 ( ∫ S 2 | F ( x , y ) | p μ 2 ( d y ) ) 1 p μ 1 ( 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 fi(y) = F(i,y) for i = 1, 2, the integral inequality gives

∥ f 1 + f 2 ∥ p = ( ∫ S 2 | ∫ S 1 F ( x , y ) μ 1 ( d x ) | p μ 2 ( d y ) ) 1 p ≤ ∫ S 1 ( ∫ S 2 | F ( x , y ) | p μ 2 ( d y ) ) 1 p μ 1 ( d x ) = ∥ f 1 ∥ p + ∥ f 2 ∥ p .

References

Minkowski inequality Wikipedia
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