Minkowski inequality


In mathematical analysis, the Minkowski inequality establishes that the Lp spaces are normed vector spaces. Let S be a measure space, let and let f and g be elements of Lp. Then is in Lp, and we have the triangle inequality
with equality for if and only if f and g are positively linearly dependent, i.e., for some or. 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. 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 numbers x1,..., xn, y1,..., yn and where n is the cardinality of S.

Proof

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 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

Suppose that and are two σ-finite measure spaces and is measurable. Then Minkowski's integral inequality is, :
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 φ and ψ.
If μ1 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
This notation has been generalized to
for, with. Using this notation, manipulation of the exponents reveals that, if, then.