Menger curvature
E199891
Menger curvature is a geometric concept that quantifies the curvature of a set or curve in metric spaces by using the reciprocal of the radius of the circle passing through three points.
All labels observed (1)
| Label | Occurrences |
|---|---|
| Menger curvature canonical | 3 |
How this entity was disambiguated
This entity first appeared as the object of triple T1780496 — 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: Menger curvature Context triple: [Karl Menger, notableWork, Menger curvature]
-
A.
Carathéodory metric
The Carathéodory metric is an intrinsic distance function in complex analysis that measures how far apart points are in a domain based on holomorphic mappings into the unit disk.
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B.
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.
-
C.
Gaussian curvature
Gaussian curvature is a fundamental concept in differential geometry that measures how a surface bends at a point by combining its principal curvatures into a single intrinsic quantity.
-
D.
Weingarten map
The Weingarten map is a differential geometric operator on a surface that encodes how the surface’s normal vector field changes, thereby describing the surface’s extrinsic curvature.
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E.
Ricci flow
Ricci flow is a geometric evolution equation that smoothly deforms the metric of a Riemannian manifold in a way analogous to heat diffusion, playing a central role in Grigori Perelman's proof of the Poincaré conjecture.
- 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: Menger curvature Target entity description: Menger curvature is a geometric concept that quantifies the curvature of a set or curve in metric spaces by using the reciprocal of the radius of the circle passing through three points.
-
A.
Carathéodory metric
The Carathéodory metric is an intrinsic distance function in complex analysis that measures how far apart points are in a domain based on holomorphic mappings into the unit disk.
-
B.
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.
-
C.
Gaussian curvature
Gaussian curvature is a fundamental concept in differential geometry that measures how a surface bends at a point by combining its principal curvatures into a single intrinsic quantity.
-
D.
Weingarten map
The Weingarten map is a differential geometric operator on a surface that encodes how the surface’s normal vector field changes, thereby describing the surface’s extrinsic curvature.
-
E.
Ricci flow
Ricci flow is a geometric evolution equation that smoothly deforms the metric of a Riemannian manifold in a way analogous to heat diffusion, playing a central role in Grigori Perelman's proof of the Poincaré conjecture.
- F. None of above. chosen
Statements (47)
| Predicate | Object |
|---|---|
| instanceOf |
curvature notion
ⓘ
geometric concept ⓘ metric geometry concept ⓘ |
| appliesTo |
curves
ⓘ
finite point configurations in Euclidean space ⓘ metric spaces ⓘ rectifiable sets ⓘ subsets of Euclidean space ⓘ |
| canBeExpressedUsing | side lengths of a triangle ⓘ |
| computationInput | three pairwise distances d(x,y), d(y,z), d(z,x) ⓘ |
| coreDefinition | reciprocal of the radius of the circle through three points ⓘ |
| definedUsing |
circumradius of a triangle
ⓘ
triples of points ⓘ |
| definitionDetail |
c(x,y,z)=0 if the three points are collinear
ⓘ
for three distinct points x,y,z, c(x,y,z)=1/R where R is the circumradius of the triangle xyz ⓘ |
| dependsOn | pairwise distances between three points ⓘ |
| dimension | has dimension of inverse length ⓘ |
| field |
analysis
ⓘ
geometric measure theory ⓘ geometry ⓘ metric geometry ⓘ |
| formulaProperty | circumradius can be computed from side lengths via Heron-type formulas ⓘ |
| generalization | can be defined in any metric space using only distances ⓘ |
| generalizes | curvature to metric spaces without differentiable structure ⓘ |
| hasIntegralVersion | integral Menger curvature ⓘ |
| historicalContext | introduced in the context of metric geometry by Karl Menger in the 20th century ⓘ |
| integralVersionDefinition | integral Menger curvature is obtained by integrating c(x,y,z)^p over triples of points ⓘ |
| integralVersionUsedIn |
quantitative descriptions of curve regularity
ⓘ
self-avoidance energies for curves ⓘ |
| invariantUnder |
Euclidean group
ⓘ
surface form:
Euclidean isometries
similarity transformations up to scaling ⓘ |
| namedAfter | Karl Menger ⓘ |
| property |
equals classical curvature for three nearby points on a smooth curve in the limit
ⓘ
nonnegative quantity ⓘ |
| relatedConcept |
Gromov’s notion of curvature in metric spaces
ⓘ
discrete curvature ⓘ second fundamental form (in smooth settings) ⓘ |
| relatedTo | classical curvature of smooth curves ⓘ |
| symbol | c(x,y,z) ⓘ |
| usedIn |
analysis of singular integrals
ⓘ
characterizations of 1-rectifiable measures ⓘ characterizations of rectifiable curves ⓘ geometric measure theory regularity results ⓘ quantitative rectifiability ⓘ study of sets of finite length ⓘ |
| usedToQuantify | how far three points deviate from being collinear ⓘ |
| zeroCondition | vanishes exactly when the three points lie on a common line ⓘ |
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: Menger curvature Description of subject: Menger curvature is a geometric concept that quantifies the curvature of a set or curve in metric spaces by using the reciprocal of the radius of the circle passing through three points.
Referenced by (3)
Full triples — surface form annotated when it differs from this entity's canonical label.
subject surface form:
Karl Menger