Minkowski inequality
E14946
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.
All labels observed (6)
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.
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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.
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C.
Kullback–Leibler divergence
Kullback–Leibler divergence is a fundamental information-theoretic measure that quantifies how one probability distribution differs from a reference distribution.
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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.
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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 for sums
ⓘ
linked to:
Minkowski inequality
Minkowski integral inequality ⓘ
linked to:
Minkowski 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
ⓘ
linked to:
Hölder 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} ⓘ
linked to:
Minkowski inequality
|
| 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
The pipeline generated the facts above by prompting gpt-5.1 with this entity's name + description and the instruction below.
Instruction
You are a knowledge base construction expert. Given a subject entity and a description of it, return factual statements that you know for the subject as a JSON list of dictionaries(triples), where keys must be "subject", "predicate" and "object". The number of facts may be very high, between 25 to 50 or more, for very popular subjects. For less popular subjects, the number of facts can be very low, like 5 or 10. # Requirements - If you don't know the subject at all, return an empty list. - If the subject is not a named entity, return an empty list. - Include at least one triple where predicate is "instanceOf". - Do not get too wordy. - Separate several objects into multiple triples with one object.
Input
Subject: Minkowski inequality Description of subject: 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.
Referenced by (14)
Full triples — surface form annotated when it differs from this entity's canonical label.
Minkowski inequality
→
statement
→
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}
ⓘ
linked to: Minkowski inequality
linked to: Minkowski inequality
linked to: Minkowski inequality
linked to: Minkowski inequality
linked to: Minkowski inequality
linked to: Minkowski inequality
linked to: Minkowski inequality