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

\|f+g\|_p \le \|f\|_p + \|g\|_p

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).

The Minkowski inequality is the triangle inequality in Lp(S).

Like Hölder's inequality, the Minkowski inequality can be specialized to sequences and vectors by using the counting measure:

\left( \sum_{k=1}^n |x_k + y_k|^p \right)^{1/p} \le \left( \sum_{k=1}^n |x_k|^p \right)^{1/p} + \left( \sum_{k=1}^n |y_k|^p \right)^{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).

[edit] Proof

First, we prove that f+g has finite p-norm if f and g both do, which follows by

|f + g|^p \le 2^{p-1}(|f|^p + |g|^p)

Indeed, here we use the fact that h(x) = xp is convex over \mathbb{R}^+ (for p greater than one) and so, if a and b are both positive then

\left(\frac{1}{2} a + \frac{1}{2} b\right)^p \le \frac{1}{2}a^p + \frac{1}{2} b^p

This means that

(a+b)^p \le 2^{p-1}a^p + 2^{p-1}b^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 Hölder's inequality

\|f + g\|_p^p = \int |f + g|^p \, \mathrm{d}\mu
\le \int (|f| + |g|)|f + g|^{p-1} \, \mathrm{d}\mu
=\int |f||f + g|^{p-1} \, \mathrm{d}\mu+\int |g||f + g|^{p-1} \, \mathrm{d}\mu
\stackrel{\text{H}\ddot{\text{o}}\text{lder}}{\le} \left( \left(\int |f|^p \, \mathrm{d}\mu\right)^{1/p} + \left (\int |g|^p \,\mathrm{d}\mu\right)^{1/p} \right) \left(\int |f + g|^{(p-1)\left(\frac{p}{p-1}\right)} \, \mathrm{d}\mu \right)^{1-\frac{1}{p}}
= (\|f\|_p + \|g\|_p)\frac{\|f + g\|_p^p}{\|f + g\|_p}

We obtain Minkowski's inequality by multiplying both sides by \frac{\|f + g\|_p}{\|f + g\|_p^p}.

[edit] References