{ "artifactId": "artifact.library.mathlib.radon-theorem.v001", "candidateOnly": false, "category": "Geometry", "claimBoundary": "This page indexes Mathlib's affine-dependence form of Radon's theorem for an indexed family in a module over the source's linearly ordered field typeclasses. From failure of affine independence it obtains a set I of indices whose image convex hull intersects the image convex hull of I's complement. The selected declaration does not assume finite dimensionality and does not directly state the familiar d+2-points-in-d-dimensions corollary. It asserts existence only, not a canonical or unique partition, a unique intersection point, or a computation.", "collectionSlug": "landmark-theorems-in-mathlib", "exactFormalStatement": "theorem Convex.radon_partition {ฮน ๐•œ E : Type*} [Field ๐•œ] [LinearOrder ๐•œ] [IsStrictOrderedRing ๐•œ] [AddCommGroup E] [Module ๐•œ E] {f : ฮน โ†’ E} (h : ยฌ AffineIndependent ๐•œ f) : โˆƒ I, (convexHull ๐•œ (f '' I) โˆฉ convexHull ๐•œ (f '' Iแถœ)).Nonempty", "family": "Convex geometry", "id": "library.mathlib.radon-theorem.v001", "links": { "collection": "/collections/landmark-theorems-in-mathlib/", "collectionData": "/data/collections/mathlib-landmarks.json", "entry": "/library-theorems/radon-theorem/", "entryData": "/data/library-theorems/radon-theorem.json", "evidence": "/proofs/artifact.library.mathlib.radon-theorem.v001.html", "evidenceData": "/data/proofs/artifact.library.mathlib.radon-theorem.v001.evidence.json", "source": "/sources/upstream/radon-theorem/" }, "landmark": { "completeFormalType": "theorem Convex.radon_partition {ฮน ๐•œ E : Type*} [Field ๐•œ] [LinearOrder ๐•œ] [IsStrictOrderedRing ๐•œ] [AddCommGroup E] [Module ๐•œ E] {f : ฮน โ†’ E} (h : ยฌ AffineIndependent ๐•œ f) : โˆƒ I, (convexHull ๐•œ (f '' I) โˆฉ convexHull ๐•œ (f '' Iแถœ)).Nonempty", "concepts": [ "affine dependence", "convex hulls", "Radon partitions", "center of mass", "convex geometry" ], "plainLanguage": "Whenever an indexed family of points has an affine dependence, its indices can be divided into a set and its complement so that the convex hulls generated by the two groups share at least one point.", "poster": { "alt": "An ivory convex-geometry poster begins with an affinely dependent indexed family, splits a finite affine relation by coefficient sign, and shows two complementary image-hull segments meeting at one point.", "byteSize": 2422965, "caption": "Affine dependence yields a complementary split of the index set whose two image convex hulls intersect.", "description": "A four-point convex-position example makes the endpoint concrete: alternating antique-gold and cobalt sign classes determine two crossing hull segments. A four-stage source-faithful route extracts an affine relation, separates coefficient signs, constructs one center of mass, and places it in both hulls, while the footer preserves the declaration's non-dimensional starting point.", "derivatives": [ { "byteSize": 85858, "height": 960, "mediaType": "image/webp", "publicPath": "/assets/library-theorems/radon-theorem/theorem-poster-v1-640.webp", "sha256": "sha256:0ee18e180ce3583ebeb23ccd5f9403850f0229ee2c2ef3f406afc08c863d4100", "width": 640 }, { "byteSize": 175262, "height": 1536, "mediaType": "image/webp", "publicPath": "/assets/library-theorems/radon-theorem/theorem-poster-v1-1024.webp", "sha256": "sha256:7c671ef3d5fff2b8657bec0b0dc5ce742212cd9afaa55b2292d1102c992bf217", "width": 1024 } ], "generatedAt": "2026-07-25T16:22:10-04:00", "height": 1536, "mediaType": "image/png", "modelRequested": "gpt-image-2", "publicPath": "/assets/library-theorems/radon-theorem/theorem-poster-v1.png", "sha256": "sha256:3b545aadb45237140027c57cb5b3914fca2416f8ae6e68a4c6b155c3b62f3e9f", "title": "Radon's Theorem at a glance", "width": 1024, "kind": "theorem_poster", "transcript": "CONVEX GEOMETRY\nRADON'S THEOREM\nAFFINE DEPENDENCE FORCES A HULL INTERSECTION\nSTART WITH AN AFFINELY DEPENDENT INDEXED FAMILY\nSPLIT THE FINITE SUPPORT BY SIGN\nTWO COMPLEMENTARY IMAGE HULLS\nMEET AT ONE POINT\nHOW THE PROOF MOVES\nEXTRACT A NONZERO AFFINE RELATION\nSEPARATE THE COEFFICIENT SIGNS\nBUILD ONE CENTER OF MASS\nPLACE IT IN BOTH CONVEX HULLS\nEXACT SCOPE\nStarts from affine dependence. No finite-dimensional or d+2-point hypothesis appears in this declaration." }, "visual": { "alt": "A text-free engraved construction separates four affinely dependent points into antique-gold and cobalt sign classes, then joins opposite same-color points by convex-hull segments that cross at one ivory point.", "byteSize": 2008372, "caption": "Splitting a nonzero affine relation by coefficient sign produces complementary image convex hulls with a common point.", "description": "Four points in convex position carry two opposed coefficient-sign colors. Their affine balance is reorganized into two complementary classes; the straight hull segment of one class crosses the straight hull segment of the other at a single highlighted point.", "derivatives": [ { "byteSize": 16680, "height": 427, "mediaType": "image/webp", "publicPath": "/assets/library-theorems/radon-theorem/theorem-schematic-v2-640.webp", "sha256": "sha256:aca79a8b52031509537d506b566f38136d8774f3aaea378689cd93800be94089", "width": 640 }, { "byteSize": 44506, "height": 800, "mediaType": "image/webp", "publicPath": "/assets/library-theorems/radon-theorem/theorem-schematic-v2-1200.webp", "sha256": "sha256:c02523d1a4128ab818ed1de7ae954a7b0890f98f067bc4caf52c6c7b90a6ef6d", "width": 1200 } ], "generatedAt": "2026-07-25T16:22:10-04:00", "height": 1024, "mediaType": "image/png", "modelRequested": "gpt-image-2", "publicPath": "/assets/library-theorems/radon-theorem/theorem-schematic-v2.png", "sha256": "sha256:2350c267fce23d0d260ea5fd9ff869ae149100b5319b5e63677bdc7045d31718", "title": "Affine dependence splits into meeting hulls", "width": 1536 }, "walkthrough": null, "whyLandmark": "Radon's theorem is a foundational partition principle in convex geometry and a key engine behind Helly-type local-to-global results. Mathlib's source proof makes the geometry explicit: an affine relation is split by coefficient sign, and the same center of mass is placed in both complementary convex hulls." }, "nonClaims": [ "ProofAtlas did not originate Radon's theorem or Mathlib's declaration.", "The selected declaration begins with affine dependence; it is not itself stated with a finite-dimensional or d+2-point hypothesis.", "The theorem does not compute a partition or common point and does not assert uniqueness or canonicity.", "The selected declaration is not Helly's theorem.", "The generated explanation and visuals are not proof evidence." ], "publicExportEligible": true, "relationship": "upstream_direct", "schemaVersion": "library-theorem-entry-data.v1", "statementId": "statement.library.mathlib.radon-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": "Every affinely dependent indexed family admits a complementary split whose two image convex hulls have a common point.", "targetId": "target.library.mathlib.radon-theorem.v001", "title": "Radon's Theorem", "upstreamOrigin": { "declarationName": "Convex.radon_partition", "immutableSourceUrl": "https://github.com/leanprover-community/mathlib4/blob/5e932f97dd25535344f80f9dd8da3aab83df0fe6/Mathlib/Analysis/Convex/Radon.lean#L46", "packageName": "mathlib", "repositoryUrl": "https://github.com/leanprover-community/mathlib4", "sourceArtifactHash": "sha256:028d32e1c80cd266e49efcee7d855bb0d1feada781c7d5ac3f044786c1b97262", "sourceByteLength": 13953, "sourceCommit": "5e932f97dd25535344f80f9dd8da3aab83df0fe6", "sourceFile": "Mathlib/Analysis/Convex/Radon.lean", "sourceLine": 46, "verificationKind": "git_worktree" }, "publicPresentationReview": { "status": "reviewed" } }