Monge problem in optimal transport
E326555
The Monge problem in optimal transport is a foundational mathematical formulation that seeks the most efficient way to move mass from one distribution to another, minimizing a given transportation cost.
All labels observed (2)
| Label | Occurrences |
|---|---|
| Monge problem in optimal transport canonical | 1 |
| Monge–Kantorovich optimal transport problem | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T3111322 — 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: Monge problem in optimal transport Context triple: [Gaspard Monge, knownFor, Monge problem in optimal transport]
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A.
Optimal Transport: Old and New
"Optimal Transport: Old and New" is a comprehensive monograph by Cédric Villani that develops the theory of optimal transport and its applications across analysis, geometry, and probability.
-
B.
Perelman’s entropy functionals
Perelman’s entropy functionals are analytic quantities introduced by Grigori Perelman to study the behavior and singularities of the Ricci flow, playing a central role in his proof of the Poincaré and geometrization conjectures.
-
C.
Carathéodory’s theorem in convex geometry
Carathéodory’s theorem in convex geometry is a fundamental result stating that any point in the convex hull of a set in ℝⁿ can be expressed as a convex combination of at most n+1 points from that set.
-
D.
Méthodes de calcul différentiel absolu et leurs applications
Méthodes de calcul différentiel absolu et leurs applications is a foundational mathematical work that systematically develops the theory of tensor calculus and its applications, laying groundwork later used in general relativity.
-
E.
Grothendieck inequality
The Grothendieck inequality is a fundamental result in functional analysis and theoretical computer science that bounds certain bilinear forms and has deep implications for Banach space theory, operator theory, and approximation algorithms.
- 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: Monge problem in optimal transport Target entity description: The Monge problem in optimal transport is a foundational mathematical formulation that seeks the most efficient way to move mass from one distribution to another, minimizing a given transportation cost.
-
A.
Optimal Transport: Old and New
"Optimal Transport: Old and New" is a comprehensive monograph by Cédric Villani that develops the theory of optimal transport and its applications across analysis, geometry, and probability.
-
B.
Perelman’s entropy functionals
Perelman’s entropy functionals are analytic quantities introduced by Grigori Perelman to study the behavior and singularities of the Ricci flow, playing a central role in his proof of the Poincaré and geometrization conjectures.
-
C.
Carathéodory’s theorem in convex geometry
Carathéodory’s theorem in convex geometry is a fundamental result stating that any point in the convex hull of a set in ℝⁿ can be expressed as a convex combination of at most n+1 points from that set.
-
D.
Méthodes de calcul différentiel absolu et leurs applications
Méthodes de calcul différentiel absolu et leurs applications is a foundational mathematical work that systematically develops the theory of tensor calculus and its applications, laying groundwork later used in general relativity.
-
E.
Grothendieck inequality
The Grothendieck inequality is a fundamental result in functional analysis and theoretical computer science that bounds certain bilinear forms and has deep implications for Banach space theory, operator theory, and approximation algorithms.
- F. None of above. chosen
Statements (48)
| Predicate | Object |
|---|---|
| instanceOf |
mathematical optimization problem
ⓘ
problem in optimal transport theory ⓘ |
| assumes | conservation of mass ⓘ |
| constraint | pushforward of source measure by T equals target measure ⓘ |
| contrastedWith | Kantorovich problem in optimal transport ⓘ |
| coreQuestion | find a transport map pushing one distribution to another with minimal cost ⓘ |
| difficulty |
may lack solutions for given marginals and cost
ⓘ
optimization problem is highly non-linear ⓘ |
| domain |
Euclidean space
ⓘ
surface form:
Euclidean spaces
Polish spaces ⓘ |
| field |
applied mathematics
ⓘ
calculus of variations ⓘ mathematical analysis ⓘ measure theory ⓘ optimal transport ⓘ probability theory ⓘ |
| formulatedBy | Gaspard Monge ⓘ |
| generalizationOf | classical earth mover problem ⓘ |
| hasConditionForExistence | absolute continuity of source measure for quadratic cost ⓘ |
| hasConditionForUniqueness | strict convexity of cost function ⓘ |
| hasContinuousVersion | Monge problem on continuous probability measures ⓘ |
| hasDiscreteVersion | Monge problem on finite point sets ⓘ |
| hasRelaxation |
Kantorovich problem in optimal transport
ⓘ
surface form:
Kantorovich formulation of optimal transport
|
| historicalSignificance | earliest formal statement of mass transportation problem ⓘ |
| inspired | development of Kantorovich duality ⓘ |
| involves |
cost function
ⓘ
source measure ⓘ target measure ⓘ transport map ⓘ |
| mathematicalFormulation | minimize integral of cost of x to T(x) over source measure ⓘ |
| namedAfter | Gaspard Monge ⓘ |
| relatedConcept |
Brenier map
ⓘ
Monge–Ampère equation ⓘ Wasserstein distance ⓘ mass transportation theory ⓘ |
| requires | deterministic transport map ⓘ |
| solutionRegularity | linked to regularity of Monge–Ampère type equations ⓘ |
| solutionType | optimal transport map ⓘ |
| typicalCostFunction |
Euclidean distance
ⓘ
general metric cost ⓘ squared Euclidean distance ⓘ |
| usedIn |
economics
ⓘ
fluid mechanics ⓘ geometry ⓘ image processing ⓘ machine learning ⓘ partial differential equations ⓘ |
| yearProposed | 1781 ⓘ |
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: Monge problem in optimal transport Description of subject: The Monge problem in optimal transport is a foundational mathematical formulation that seeks the most efficient way to move mass from one distribution to another, minimizing a given transportation cost.
Referenced by (2)
Full triples — surface form annotated when it differs from this entity's canonical label.
this entity surface form:
Monge–Kantorovich optimal transport problem