Minkowski inequality

This page is about Minkowski's inequality for norms. See Minkowski's first inequality for convex bodies for Minkowski's inequality in convex geometry.
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 (which means f = λ g or g = λ f for some λ ≥ 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 righthand 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).
Contents
Proof
First, we prove that f+g has finite pnorm if f and g both do, which follows by
Indeed, here we use the fact that h(x) = x^{p} is convex over (for p greater than one) and so, if a and b are both positive then, by Jensen's inequality,
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 Hölder's inequality
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 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 nonnegative measurable functions φ and ψ.
If μ_{1} is the counting measure on a twopoint set S_{1} = {1,2}, then Minkowski's integral inequality gives the usual Minkowski inequality as a special case: for putting ƒ_{i}(y) = F(i,y) for i = 1,2, the integral inequality gives
See also
References
 Hardy, G. H.; Littlewood, J. E.; Pólya, G. (1952). Inequalities. Cambridge Mathematical Library (second ed.). Cambridge: Cambridge University Press. ISBN 0521358809.
 Minkowski, H. (1953). Geometrie der Zahlen. Chelsea.
 Stein, Elias (1970). Singular integrals and differentiability properties of functions. Princeton University Press.
 M.I. Voitsekhovskii (2001), "Minkowski inequality", in Hazewinkel, Michiel, Encyclopaedia of Mathematics, Springer, ISBN 9781556080104, http://eom.springer.de/M/m064060.htm
 Arthur Lohwater (1982). "Introduction to Inequalities". Online ebook in PDF format. http://www.mediafire.com/?1mw1tkgozzu.
Categories: Inequalities
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