{
  "artifactId": "artifact.library.mathlib.helly-theorem.v001",
  "candidateOnly": false,
  "category": "Geometry",
  "claimBoundary": "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.",
  "collectionSlug": "landmark-theorems-in-mathlib",
  "exactFormalStatement": "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",
  "family": "Convex geometry",
  "id": "library.mathlib.helly-theorem.v001",
  "links": {
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    "collectionData": "/data/collections/mathlib-landmarks.json",
    "entry": "/library-theorems/helly-theorem/",
    "entryData": "/data/library-theorems/helly-theorem.json",
    "evidence": "/proofs/artifact.library.mathlib.helly-theorem.v001.html",
    "evidenceData": "/data/proofs/artifact.library.mathlib.helly-theorem.v001.evidence.json",
    "source": "/sources/upstream/helly-theorem/"
  },
  "landmark": {
    "completeFormalType": "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",
    "concepts": [
      "convex sets",
      "finite-dimensional spaces",
      "common intersections",
      "Radon partitions",
      "affine dependence"
    ],
    "plainLanguage": "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.",
    "poster": {
      "alt": "An editorial theorem poster explains the finite d-dimensional intersection theorem and its Radon-partition proof route.",
      "byteSize": 1958602,
      "caption": "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.",
      "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.",
      "derivatives": [
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          "height": 960,
          "mediaType": "image/webp",
          "publicPath": "/assets/library-theorems/helly-theorem/theorem-poster-v2-640.webp",
          "sha256": "sha256:3b1ca101bab32b398b86d2f77a050d296f2e49c002ae780ed1e6e54614a0a94e",
          "width": 640
        },
        {
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          "height": 1536,
          "mediaType": "image/webp",
          "publicPath": "/assets/library-theorems/helly-theorem/theorem-poster-v2-1024.webp",
          "sha256": "sha256:bd04bb4adf5113df6e111299f89f90c22ff57ba721bd1e4815887674f2a2b174",
          "width": 1024
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      ],
      "generatedAt": "2026-07-25T12:05:00.000Z",
      "height": 1536,
      "mediaType": "image/png",
      "modelRequested": "gpt-image-2",
      "publicPath": "/assets/library-theorems/helly-theorem/theorem-poster-v2.png",
      "sha256": "sha256:9324ac8346d5be7360abf02341a156d5b1e1d929ac2f7a970dab0576ae153d40",
      "title": "Helly’s Theorem at a glance",
      "width": 1024,
      "kind": "theorem_poster",
      "transcript": "CONVEX GEOMETRY · FINITE DIMENSION\nHELLY’S THEOREM\nCHECK d+1 AT A TIME\nIN DIMENSION d\nAT LEAST d+1 INDEXED SETS\nEVERY d+1 MEET\n→\nTHE WHOLE FINITE FAMILY MEETS\nCHOOSE OMIT-ONE WITNESSES\nFORCE AFFINE DEPENDENCE\nAPPLY RADON’S PARTITION\nRECOVER THE COMMON POINT\nEXACT SCOPE\nFinite indexed convex family. Exact conclusion: nonempty total intersection."
    },
    "visual": {
      "alt": "A planar example of convex sets sharing one point is paired with omit-one witnesses and a two-color Radon partition whose hulls meet.",
      "byteSize": 2020810,
      "caption": "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.",
      "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.",
      "derivatives": [
        {
          "byteSize": 29344,
          "height": 427,
          "mediaType": "image/webp",
          "publicPath": "/assets/library-theorems/helly-theorem/theorem-schematic-v2-640.webp",
          "sha256": "sha256:b65051a08bb97c442b569f2609038c2af25f02a6e181634a15f0cf6523d30448",
          "width": 640
        },
        {
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          "height": 800,
          "mediaType": "image/webp",
          "publicPath": "/assets/library-theorems/helly-theorem/theorem-schematic-v2-1200.webp",
          "sha256": "sha256:2087d40d3db2718f3762764fee665dbcd76acc5bfcacedfbd20d6f83982b4abd",
          "width": 1200
        }
      ],
      "generatedAt": "2026-07-25T12:05:00.000Z",
      "height": 1024,
      "mediaType": "image/png",
      "modelRequested": "gpt-image-2",
      "publicPath": "/assets/library-theorems/helly-theorem/theorem-schematic-v2.png",
      "sha256": "sha256:ac29ad001190a30723b9c9357531927e1f83d71742017ba6363552d72fab554a",
      "title": "Local convex intersections force a common point",
      "width": 1536
    },
    "walkthrough": null,
    "whyLandmark": "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."
  },
  "nonClaims": [
    "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."
  ],
  "publicExportEligible": true,
  "relationship": "upstream_direct",
  "schemaVersion": "library-theorem-entry-data.v1",
  "statementId": "statement.library.mathlib.helly-theorem.v001",
  "statusAxes": {
    "acceptedAtlasResult": "Not recorded for the preferred artifact",
    "locallyReproduced": "Exact upstream declaration replayed",
    "reviewedPage": "Current public presentation reviewed",
    "upstreamIndexed": "Pinned source bytes verified locally"
  },
  "summary": "For a sufficiently large finite family of convex sets in dimension d, nonempty intersection of every d+1 indexed members forces a nonempty total intersection.",
  "targetId": "target.library.mathlib.helly-theorem.v001",
  "title": "Helly’s Theorem",
  "upstreamOrigin": {
    "declarationName": "Convex.helly_theorem",
    "immutableSourceUrl": "https://github.com/leanprover-community/mathlib4/blob/5e932f97dd25535344f80f9dd8da3aab83df0fe6/Mathlib/Analysis/Convex/Radon.lean#L143",
    "packageName": "mathlib",
    "repositoryUrl": "https://github.com/leanprover-community/mathlib4",
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    "sourceFile": "Mathlib/Analysis/Convex/Radon.lean",
    "sourceLine": 143,
    "verificationKind": "git_worktree"
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  "publicPresentationReview": {
    "status": "reviewed"
  }
}