Minkowski functional
E14948
The Minkowski functional is a mathematical tool in functional analysis that assigns a nonnegative real number to each vector in a vector space based on its position relative to a given convex, balanced, absorbing set, generalizing the notion of a norm.
All labels observed (1)
| Label | Occurrences |
|---|---|
| Minkowski functional canonical | 3 |
How this entity was disambiguated
This entity first appeared as the object of triple T130827 — 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 functional Context triple: [Hermann Minkowski, knownFor, Minkowski functional]
-
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.
Kakutani fixed-point theorem
The Kakutani fixed-point theorem is a fundamental result in mathematical analysis and game theory that guarantees the existence of fixed points for certain set-valued (multivalued) functions, underpinning key existence proofs such as Nash equilibria.
-
C.
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.
-
D.
Nash embedding theorem
The Nash embedding theorem is a fundamental result in differential geometry that shows any Riemannian manifold can be isometrically embedded into some Euclidean space, thereby realizing abstract curved spaces as concrete subsets of standard Euclidean space.
-
E.
implicit function theorem
The implicit function theorem is a fundamental result in calculus and differential geometry that guarantees, under suitable smoothness and nondegeneracy conditions, the local solvability of equations for some variables as differentiable functions of others.
- 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 functional Target entity description: The Minkowski functional is a mathematical tool in functional analysis that assigns a nonnegative real number to each vector in a vector space based on its position relative to a given convex, balanced, absorbing set, generalizing the notion of a norm.
-
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.
Kakutani fixed-point theorem
The Kakutani fixed-point theorem is a fundamental result in mathematical analysis and game theory that guarantees the existence of fixed points for certain set-valued (multivalued) functions, underpinning key existence proofs such as Nash equilibria.
-
C.
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.
-
D.
Nash embedding theorem
The Nash embedding theorem is a fundamental result in differential geometry that shows any Riemannian manifold can be isometrically embedded into some Euclidean space, thereby realizing abstract curved spaces as concrete subsets of standard Euclidean space.
-
E.
implicit function theorem
The implicit function theorem is a fundamental result in calculus and differential geometry that guarantees, under suitable smoothness and nondegeneracy conditions, the local solvability of equations for some variables as differentiable functions of others.
- F. None of above. chosen
Statements (44)
| Predicate | Object |
|---|---|
| instanceOf |
functional
ⓘ
mathematical concept ⓘ |
| alsoKnownAs |
gauge
ⓘ
gauge functional ⓘ |
| application |
characterizing bounded sets in locally convex spaces
ⓘ
defining metrics on topological vector spaces ⓘ formulating Hahn–Banach type results ⓘ |
| associatesToEachSet | a sublinear functional ⓘ |
| category |
functional-analytic construction
ⓘ
tool in convex geometry ⓘ |
| codomain | nonnegative real numbers ⓘ |
| conditionForNorm |
set is absorbing
ⓘ
set is balanced ⓘ set is bounded ⓘ set is closed ⓘ set is convex ⓘ set is symmetric about the origin ⓘ |
| definedUsing |
absorbing set
ⓘ
balanced set ⓘ convex set ⓘ |
| dependsOn | choice of convex balanced absorbing set ⓘ |
| domain | vector space ⓘ |
| field |
convex analysis
ⓘ
functional analysis ⓘ |
| generalizes | norm ⓘ |
| input | vector ⓘ |
| invariantUnder | scaling of the defining set by positive constants up to equivalence ⓘ |
| namedAfter | Hermann Minkowski ⓘ |
| output | nonnegative scalar ⓘ |
| property |
nonnegative
ⓘ
positively homogeneous ⓘ subadditive ⓘ sublinear ⓘ vanishes at the origin ⓘ |
| relatedTo |
gauge of a convex set
ⓘ
seminorm ⓘ support function ⓘ |
| typicalAssumptionOnSet | contains the origin ⓘ |
| usedIn |
construction of locally convex topologies
ⓘ
definition of seminorms ⓘ duality theory in convex analysis ⓘ study of convex bodies ⓘ theory of locally convex spaces ⓘ |
| yields | norm on the linear span of the set under suitable conditions ⓘ |
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 functional Description of subject: The Minkowski functional is a mathematical tool in functional analysis that assigns a nonnegative real number to each vector in a vector space based on its position relative to a given convex, balanced, absorbing set, generalizing the notion of a norm.
Referenced by (3)
Full triples — surface form annotated when it differs from this entity's canonical label.