Triple

Given a phrase, classify it is english named entity (e.g., persons, organizations, works of art) in Latin script, or not (e.g., literals, dates, URLs, verbose phrases). For disambiguation, the statement where the phrase occurs as object is also given. Please return a JSON object with `phrase` (string, the phrase being analyzed) and `is_ne` (boolean, indicating whether the phrase is a Named Entity).

Phrase: fractional Helly theorem | Statement: [Helly’s theorem, relatedTo, fractional Helly theorem]

Target entity: fractional Helly theorem
Context triple: [Helly’s theorem, relatedTo, fractional Helly theorem]

• A. Helly’s theorem
  Helly’s theorem is a fundamental result in convex geometry that gives conditions under which a family of convex sets in Euclidean space has a nonempty common intersection.
• 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. Radon’s theorem
  Radon’s theorem is a fundamental result in convex geometry stating that any set of sufficiently many points in Euclidean space can be partitioned into two disjoint subsets whose convex hulls intersect.
• D. Borsuk’s conjecture in geometry
  Borsuk’s conjecture in geometry is a famous (now disproven in higher dimensions) problem in metric geometry that proposed any bounded set in n-dimensional Euclidean space can be partitioned into n+1 subsets of smaller diameter.
• E. Danzer set in discrete geometry
  A Danzer set in discrete geometry is a hypothetical point set in Euclidean space that intersects every convex body of a given volume while maintaining uniformly bounded density.
• F. None of above. chosen
• G. Unsure - the case is ambiguous/there is not enough information to decide.

Target entity: fractional Helly theorem
Target entity description: The fractional Helly theorem is a result in combinatorial geometry that generalizes Helly’s theorem by asserting that if a sufficiently large fraction of small subfamilies of convex sets intersect, then a large subfamily of the whole collection has a common intersection.

• A. Helly’s theorem
  Helly’s theorem is a fundamental result in convex geometry that gives conditions under which a family of convex sets in Euclidean space has a nonempty common intersection.
• 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. Radon’s theorem
  Radon’s theorem is a fundamental result in convex geometry stating that any set of sufficiently many points in Euclidean space can be partitioned into two disjoint subsets whose convex hulls intersect.
• D. Borsuk’s conjecture in geometry
  Borsuk’s conjecture in geometry is a famous (now disproven in higher dimensions) problem in metric geometry that proposed any bounded set in n-dimensional Euclidean space can be partitioned into n+1 subsets of smaller diameter.
• E. Danzer set in discrete geometry
  A Danzer set in discrete geometry is a hypothetical point set in Euclidean space that intersects every convex body of a given volume while maintaining uniformly bounded density.
• F. None of above. chosen