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