Geometry · existing Mathlib declaration
Helly’s Theorem
In a finite-dimensional space, consider a finite indexed family of convex sets. Once the family has at least d+1 members, it is enough to know that every choice of exactly d+1 sets meets: then all the sets share a point.
- convex sets
- finite-dimensional spaces
- common intersections
- Radon partitions
- affine dependence
The theorem at a glance
Helly’s Theorem at a glance
A source-bound editorial overview of the exact Mathlib formulation. The image is explanatory; the statement map, formal type, and pinned source remain authoritative.
Helly’s Theorem at a glance
Detailed visual description
A planar window makes the local-to-global statement intuitive. A connected proof ribbon then chooses omit-one witnesses, forces affine dependence, applies Radon’s partition, and recovers a point in every original convex set.
Formal orientation
Statement map
This deterministic map orients the reader from the mathematical summary to the pinned declaration and exact checked form. The HTML statement below is authoritative.
Local convex intersections force a common point
Detailed visual description
A labelled planar example illustrates the endpoint while a second geometric route shows omit-one witnesses, affine dependence, a Radon split, and the shared point recovered by convexity.
Complementary intuition
A mathematical landmark
Helly’s theorem is a central local-to-global principle in convex geometry. Mathlib derives the finite theorem through Radon’s partition theorem and affine dependence, providing a clear geometric proof architecture.
The low-text schematic gives the theorem a visual identity; it does not replace the source-bound poster or exact statement.
Authoritative formal type
Exact Mathlib statement
theorem Convex.helly_theorem {ι 𝕜 E : Type*} [Field 𝕜] [LinearOrder 𝕜] [IsStrictOrderedRing 𝕜] [AddCommGroup E] [Module 𝕜 E] [FiniteDimensional 𝕜 E] {F : ι → Set E} {s : Finset ι} (h_card : Module.finrank 𝕜 E + 1 ≤ s.card) (h_convex : ∀ i ∈ s, Convex 𝕜 (F i)) (h_inter : ∀ I ⊆ s, I.card = Module.finrank 𝕜 E + 1 → (⋂ i ∈ I, F i).Nonempty) : (⋂ i ∈ s, F i).NonemptySelected evidence declaration text
theorem Convex.helly_theorem {ι 𝕜 E : Type*} [Field 𝕜] [LinearOrder 𝕜] [IsStrictOrderedRing 𝕜] [AddCommGroup E] [Module 𝕜 E] [FiniteDimensional 𝕜 E] {F : ι → Set E} {s : Finset ι} (h_card : Module.finrank 𝕜 E + 1 ≤ s.card) (h_convex : ∀ i ∈ s, Convex 𝕜 (F i)) (h_inter : ∀ I ⊆ s, I.card = Module.finrank 𝕜 E + 1 → (⋂ i ∈ I, F i).Nonempty) : (⋂ i ∈ s, F i).NonemptyProofAtlas record
What has been checked
These states distinguish upstream identity, local reproduction, review, and Atlas acceptance. This page is part of the public, read-only Mathlib landmark collection.
Claim boundary
No new theorem is claimed
This page indexes Mathlib’s finite indexed-family Helly theorem in a finite-dimensional module over the source’s linearly ordered field typeclasses. The family has at least d+1 indices, and every subfamily of exactly d+1 indices must intersect. The selected endpoint proves nonemptiness only; it is not the separate compact infinite-family variant, an algorithm, an explicit witness construction, or an optimality theorem. Planar diagrams are examples, not the theorem’s full scope.
- ProofAtlas did not originate Helly’s theorem or Mathlib’s declaration.
- The selected endpoint does not assert the unrestricted infinite-family theorem.
- It does not compute a common point or prove the optimality of d+1.
- The generated explanation and visuals are not proof evidence.
Provenance
Origin and evidence stay separate
- Existing declaration
Convex.helly_theoremin mathlib- Relationship
- The checked artifact is the upstream declaration itself
- Preferred checked artifact
artifact.library.mathlib.helly-theorem.v001- Source handling
- Verified upstream reference; no mirrored source package