Lempert function on convex domains
E521040
The Lempert function on convex domains is a complex-analytic invariant that coincides with the Kobayashi distance and provides an extremal characterization of holomorphic mappings between convex domains in several complex variables.
All labels observed (1)
| Label | Occurrences |
|---|---|
| Lempert function on convex domains canonical | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T5446381 — 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: Lempert function on convex domains Context triple: [Carathéodory metric, isUpperBoundFor, Lempert function on convex domains]
-
A.
Nevanlinna–Pick interpolation
Nevanlinna–Pick interpolation is a classical problem in complex analysis and operator theory that seeks analytic functions, typically bounded by one in the unit disk, which match prescribed values at given points.
-
B.
Inequalities for analytic functions
"Inequalities for analytic functions" is a mathematical work by Gábor Szegő that develops fundamental bounds and estimates for complex analytic functions, particularly in the context of complex analysis and approximation theory.
-
C.
Szegő kernel
The Szegő kernel is a fundamental reproducing kernel in complex analysis and operator theory, associated with Hardy spaces on the boundary of a domain and central to the study of orthogonal polynomials and boundary behavior of analytic functions.
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D.
Dynamics in One Complex Variable
Dynamics in One Complex Variable is a foundational graduate-level textbook by John Milnor that introduces and develops the theory of complex dynamical systems, particularly the iteration of rational maps on the Riemann sphere.
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E.
Runge approximation theorem
The Runge approximation theorem is a fundamental result in complex analysis stating that holomorphic functions on certain domains can be uniformly approximated by rational functions with poles outside those domains.
- 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: Lempert function on convex domains Target entity description: The Lempert function on convex domains is a complex-analytic invariant that coincides with the Kobayashi distance and provides an extremal characterization of holomorphic mappings between convex domains in several complex variables.
-
A.
Nevanlinna–Pick interpolation
Nevanlinna–Pick interpolation is a classical problem in complex analysis and operator theory that seeks analytic functions, typically bounded by one in the unit disk, which match prescribed values at given points.
-
B.
Inequalities for analytic functions
"Inequalities for analytic functions" is a mathematical work by Gábor Szegő that develops fundamental bounds and estimates for complex analytic functions, particularly in the context of complex analysis and approximation theory.
-
C.
Szegő kernel
The Szegő kernel is a fundamental reproducing kernel in complex analysis and operator theory, associated with Hardy spaces on the boundary of a domain and central to the study of orthogonal polynomials and boundary behavior of analytic functions.
-
D.
Dynamics in One Complex Variable
Dynamics in One Complex Variable is a foundational graduate-level textbook by John Milnor that introduces and develops the theory of complex dynamical systems, particularly the iteration of rational maps on the Riemann sphere.
-
E.
Runge approximation theorem
The Runge approximation theorem is a fundamental result in complex analysis stating that holomorphic functions on certain domains can be uniformly approximated by rational functions with poles outside those domains.
- F. None of above. chosen
Statements (46)
| Predicate | Object |
|---|---|
| instanceOf |
complex-analytic invariant
ⓘ
distance-like function ⓘ holomorphic invariant ⓘ |
| appearsIn | Lempert’s 1981–1982 work on complex geodesics in convex domains ⓘ |
| appliesTo |
bounded convex domains
ⓘ
unbounded convex domains ⓘ |
| associatedWith | extremal discs ⓘ |
| characterizes | extremal holomorphic mappings between convex domains ⓘ |
| coincidesWith |
Kobayashi distance on convex domains
ⓘ
integrated Kobayashi distance on convex domains ⓘ |
| definedOn |
convex domains in C^n
ⓘ
domains in several complex variables ⓘ |
| definedVia |
extremal holomorphic discs
ⓘ
infimum over holomorphic discs joining two points ⓘ |
| dependsOn | complex structure of the domain ⓘ |
| generalizes | Poincaré distance on the unit disc ⓘ |
| hasProperty |
biholomorphic invariance
ⓘ
monotonicity with respect to domain inclusion ⓘ |
| hasSpecialCase |
Lempert function on convex balanced domains
ⓘ
Lempert function on strictly convex domains ⓘ Lempert function on the unit ball in C^n ⓘ |
| introducedBy | László Lempert ⓘ |
| invariantUnder |
biholomorphic mappings
ⓘ
holomorphic automorphisms of the domain ⓘ |
| is | holomorphically contractible invariant ⓘ |
| isToolFor |
studying biholomorphic equivalence of convex domains
ⓘ
studying extremal problems for holomorphic maps ⓘ |
| playsRoleIn |
equivalence of invariant metrics on convex domains
ⓘ
theory of complex geodesics in convex domains ⓘ |
| relatedTo |
Carathéodory distance
ⓘ
linked to:
Carathéodory metric
Kobayashi distance ⓘ Schwarz–Pick type inequalities in several variables ⓘ complex geodesics ⓘ holomorphic retracts of convex domains ⓘ intrinsic pseudodistances ⓘ |
| satisfies | triangle inequality ⓘ |
| studiedIn |
complex analysis in several variables
ⓘ
geometric function theory in higher dimensions ⓘ |
| takesValuesIn | [0,+∞] ⓘ |
| usedIn |
complex geometry
ⓘ
invariant metrics on complex domains ⓘ several complex variables ⓘ |
| usedToCharacterize | complex geodesics in convex domains ⓘ |
| usedToProve | equivalence of Kobayashi and Carathéodory distances on certain convex domains ⓘ |
| usedToStudy |
holomorphic mappings between convex domains
ⓘ
intrinsic geometry of convex domains ⓘ |
How these facts were elicited
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Instruction
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Input
Subject: Lempert function on convex domains Description of subject: The Lempert function on convex domains is a complex-analytic invariant that coincides with the Kobayashi distance and provides an extremal characterization of holomorphic mappings between convex domains in several complex variables.
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.