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.

Try in SPARQL Jump to: Surface forms Disambiguation Statements Elicitation Referenced by

All labels observed (5)

Label	Occurrences
Minkowski inequality canonical	7
Minkowski integral inequality	2
Minkowski's inequality	2
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}	1
Minkowski inequality for sums	1

How this entity was disambiguated

This entity first appeared as the object of triple T130825 — resolving that mention is where its identity was fixed. The disambiguator weighed these candidate entities and picked the highlighted one (or “None”, minting a new entity). This is how homonymy is resolved: the same surface form can point to different entities.

NED1 Entity disambiguation (via context triple) gpt-5-mini-2025-08-07

Target entity: Minkowski inequality
Context triple: [Hermann Minkowski, knownFor, Minkowski inequality]

A. Hilbert spaces
Hilbert spaces are complete inner product spaces that provide the fundamental framework for modern functional analysis and many areas of mathematical physics.
B. Rényi entropy
Rényi entropy is a generalized measure of information and uncertainty that extends Shannon entropy by introducing a tunable order parameter to emphasize different aspects of a probability distribution.
C. Kullback–Leibler divergence
Kullback–Leibler divergence is a fundamental information-theoretic measure that quantifies how one probability distribution differs from a reference distribution.
D. Feynman–Kac formula
The Feynman–Kac formula is a fundamental result connecting solutions of certain partial differential equations with expectations over stochastic processes, forming a bridge between quantum mechanics, probability theory, and mathematical finance.
E. Feynman–Hellmann theorem
The Feynman–Hellmann theorem is a result in quantum mechanics that relates the derivative of an energy eigenvalue with respect to a parameter in the Hamiltonian to the expectation value of the corresponding derivative of the Hamiltonian.
F. None of above. chosen
G. Unsure - the case is ambiguous/there is not enough information to decide.

NED2 Entity disambiguation (via description) gpt-5-mini-2025-08-07

Target entity: Minkowski inequality
Target entity description: 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.

A. Hilbert spaces
Hilbert spaces are complete inner product spaces that provide the fundamental framework for modern functional analysis and many areas of mathematical physics.
B. Rényi entropy
Rényi entropy is a generalized measure of information and uncertainty that extends Shannon entropy by introducing a tunable order parameter to emphasize different aspects of a probability distribution.
C. Kullback–Leibler divergence
Kullback–Leibler divergence is a fundamental information-theoretic measure that quantifies how one probability distribution differs from a reference distribution.
D. Feynman–Kac formula
The Feynman–Kac formula is a fundamental result connecting solutions of certain partial differential equations with expectations over stochastic processes, forming a bridge between quantum mechanics, probability theory, and mathematical finance.
E. Feynman–Hellmann theorem
The Feynman–Hellmann theorem is a result in quantum mechanics that relates the derivative of an energy eigenvalue with respect to a parameter in the Hamiltonian to the expectation value of the corresponding derivative of the Hamiltonian.
F. None of above. chosen

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 self-linksurface differs ⓘ surface form: Minkowski inequality for sums Minkowski inequality self-linksurface differs ⓘ surface form: 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	Hölder inequality ⓘ surface form: 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 ⓘ Minkowski inequality self-linksurface differs ⓘ surface form: 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 ⓘ

How these facts were elicited

Referenced by (13)

Full triples — surface form annotated when it differs from this entity's canonical label.

Hermann Minkowski → knownFor → Minkowski inequality ⓘ

Minkowski inequality → statement → Minkowski inequality self-linksurface differs ⓘ

this entity surface form: 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}

Minkowski inequality → hasVariant → Minkowski inequality self-linksurface differs ⓘ

this entity surface form: Minkowski integral inequality

Minkowski inequality → hasVariant → Minkowski inequality self-linksurface differs ⓘ

this entity surface form: Minkowski inequality for sums

Hermann → knownFor → Minkowski inequality ⓘ

subject surface form: Hermann Minkowski

Hölder inequality → implies → Minkowski inequality ⓘ

Hölder inequality → relatedTo → Minkowski inequality ⓘ

Inequalities → containsTopic → Minkowski inequality ⓘ

Cauchy–Schwarz inequality → relatedTo → Minkowski inequality ⓘ

Young inequality for convolutions → involvesConcept → Minkowski inequality ⓘ

Young inequality for convolutions → proofUses → Minkowski inequality ⓘ

this entity surface form: Minkowski integral inequality

Young's inequality → isRelatedTo → Minkowski inequality ⓘ

this entity surface form: Minkowski's inequality

Young's inequality → isUsedToProve → Minkowski inequality ⓘ