Minkowski inequality

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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

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

with equality for 1<p<∞ only if f and g are linearly dependent.

The Minkowski inequality is the triangle inequality in L^p(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, ..., x_n, y₁, ..., y_n and where n is the cardinality of S.

[edit] Proof

Choose q so that 1/p + 1/q =1. Then

$\int |f+g|^p d\mu$

$\le\int |f+g|^{p-1}|f| d\mu+\int |f+g|^{p-1}|g| d\mu$ (because |f+g| ≤ |f|+|g|)

$\le\left(\int |f+g|^{(p-1)q} d\mu\right)^{1/q}\left(\int |f|^p d\mu\right)^{1/p} + \left(\int |f+g|^{(p-1)q} d\mu\right)^{1/q}\left(\int |g|^p d\mu\right)^{1/p}$ (by Hölder's inequality).

Since (p-1)q = p and 1/q = 1-1/p, we obtain Minkowski's inequality by multiplying both sides by ∫(|f+g|^pdμ)^(1/p)−1.

[edit] References

G.H. Hardy, J.E. Littlewood, G. Pólya, Inequalities , Cambridge Univ. Press (1934) ISBN 0-521-35880-9
H. Minkowski, Geometrie der Zahlen , Chelsea, reprint (1953)
M.I. Voitsekhovskii, "Minkowski inequality" SpringerLink Encyclopaedia of Mathematics (2001)

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Category: Inequalities