Minkowski inequality

E14946

mathematical inequality theorem

The Minkowski inequality is a fundamental result in functional analysis and measure theory that generalizes the triangle inequality to L^p spaces, providing a key tool for studying norms and integrable functions.

Aliases (3)

Statements (47)

Predicate	Object
instanceOf	mathematical inequality → theorem →
appliesTo	L^p norm → Lp space →
assumes	underlying measure space is σ-finite (in many standard formulations) →
category	inequalities in analysis →
dependsOn	Hölder inequality →
domain	complex-valued functions → real-valued functions →
ensures	Lp is a metric space → Lp norm is subadditive →
expresses	convexity of the L^p norm →
field	functional analysis → mathematical analysis → measure theory →
generalizes	triangle inequality →
hasConsequence	stability of L^p norms under addition →
hasVariant	Minkowski inequality for sums → Minkowski integral inequality →
holdsIn	Lebesgue spaces → finite-dimensional Euclidean spaces → sequence spaces ℓ^p →
implies	L^p norm satisfies triangle inequality →
introducedIn	early 20th century →
isSpecialCaseOf	Khinchin–Kahane type inequalities →
mathematicalDomain	Banach spaces → normed vector spaces →
namedAfter	Hermann Minkowski →
relatedTo	Cauchy–Schwarz inequality → Hölder inequality → Jensen inequality →
requires	1 ≤ p ≤ ∞ →
statement	For 1 ≤ p < ∞ and measurable functions f,g with finite L^p norms, \|\|f+g\|\|_p ≤ \|\|f\|\|_p + \|\|g\|\|_p → For sequences (x_k),(y_k) in ℓ^p, (∑\|x_k + y_k\|^p)^{1/p} ≤ (∑\|x_k\|^p)^{1/p} + (∑\|y_k\|^p)^{1/p} →
type	norm inequality →
usedFor	analysis of integrable functions → establishing completeness of L^p spaces → establishing inequalities between norms → proving that L^p is a normed space →
usedIn	Fourier analysis → partial differential equations → probability theory → signal processing → statistics →
usedToShow	Lp convergence properties → Lp spaces are convex →
validFor	measurable functions with finite p-th power integral →

Referenced by (7)

Subject (surface form when different)	Predicate
Minkowski inequality ("Minkowski integral inequality") → Minkowski inequality ("Minkowski inequality for sums") →	hasVariant
Hermann Minkowski → Hermann Minkowski →	knownFor
Hölder inequality →	implies
Hölder inequality →	relatedTo
Minkowski inequality ("For sequences (x_k),(y_k) in ℓ^p, (∑\|x_k + y_k\|^p)^{1/p} ≤ (∑\|x_k\|^p)^{1/p} + (∑\|y_k\|^p)^{1/p}") →	statement