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
with equality for 1 < p < ∞ if and only if f and g are positively linearly dependent (which means f = g or g = f for some ≥ 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 Lp(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 x1, ..., xn, y1, ..., yn and where n is the cardinality of S (the number of elements in S).
Contents |
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 (for 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
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):
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 ƒi(y) = F(i,y) for i = 1,2, the integral inequality gives