Existing in Mathlib · Geometry

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

Exact theorem

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).Nonempty

The theorem at a glance

Helly’s Theorem at a glance

A visual guide to the theorem's hypotheses, structure, and conclusion; the exact statement gives the formal detail.

An editorial theorem poster explains the finite d-dimensional intersection theorem and its Radon-partition proof route. — For a finite indexed convex family in dimension d with at least d+1 members, intersections of every d+1 members determine one global intersection.

Statement structure

From hypotheses to conclusion

This map summarizes the statement's structure; the exact Mathlib declaration remains authoritative.

A planar example of convex sets sharing one point is paired with omit-one witnesses and a two-color Radon partition whose hulls meet. — In dimension d, once a finite indexed convex family has at least d+1 members, checking every d+1 sets forces a common intersection point.

Why it matters

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.

ProofAtlas record

What has been checked

Upstream indexedPinned source bytes verified locally

Locally reproducedExact upstream declaration replayed

Reviewed pageCurrent public presentation reviewed

Accepted Atlas resultNot recorded for the preferred artifact

Mathlib is the source of the theorem; the local Lean replay and page review are separate.

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.

Source and local evidence

Where the theorem comes from

Existing declaration: Convex.helly_theorem in mathlib
Relationship: The checked artifact is the upstream declaration itself
Local Lean evidence: artifact.library.mathlib.helly-theorem.v001
Source: Open the pinned upstream reference