{ "artifactId": "artifact.library.mathlib.krein-milman-theorem.v001", "candidateOnly": false, "category": "Analysis", "claimBoundary": "This page indexes Mathlib's Krein–Milman theorem in a real Hausdorff locally convex topological vector space: every compact convex set s equals closure (convexHull ℝ (s.extremePoints ℝ)). No finite-dimensional hypothesis is present, and the closure is part of the formal conclusion. The set may be empty. The selected declaration does not supply a finite convex representation, a bound on the number of extreme points, or the finite-dimensional strengthening in which closure can sometimes be removed.", "collectionSlug": "landmark-theorems-in-mathlib", "exactFormalStatement": "theorem closure_convexHull_extremePoints {E : Type*} [AddCommGroup E] [Module ℝ E] [TopologicalSpace E] [T2Space E] [IsTopologicalAddGroup E] [ContinuousSMul ℝ E] [LocallyConvexSpace ℝ E] {s : Set E} (hscomp : IsCompact s) (hAconv : Convex ℝ s) : closure (convexHull ℝ (s.extremePoints ℝ)) = s", "family": "Convexity in locally convex spaces", "id": "library.mathlib.krein-milman-theorem.v001", "links": { "collection": "/collections/landmark-theorems-in-mathlib/", "collectionData": "/data/collections/mathlib-landmarks.json", "entry": "/library-theorems/krein-milman-theorem/", "entryData": "/data/library-theorems/krein-milman-theorem.json", "evidence": "/proofs/artifact.library.mathlib.krein-milman-theorem.v001.html", "evidenceData": "/data/proofs/artifact.library.mathlib.krein-milman-theorem.v001.evidence.json", "source": "/sources/upstream/krein-milman-theorem/" }, "landmark": { "completeFormalType": "theorem closure_convexHull_extremePoints {E : Type*} [AddCommGroup E] [Module ℝ E] [TopologicalSpace E] [T2Space E] [IsTopologicalAddGroup E] [ContinuousSMul ℝ E] [LocallyConvexSpace ℝ E] {s : Set E} (hscomp : IsCompact s) (hAconv : Convex ℝ s) : closure (convexHull ℝ (s.extremePoints ℝ)) = s", "concepts": [ "compact convex sets", "extreme points", "closed convex hull", "locally convex spaces", "geometric Hahn–Banach separation", "exposed faces" ], "plainLanguage": "In a real Hausdorff locally convex topological vector space, a compact convex set is completely recovered by taking all convex combinations of its extreme points and then taking the closure. The closure matters: this statement does not reduce the set to a finite polytope or promise a finite representation of each point.", "poster": { "alt": "An ivory convex-analysis poster states that a compact convex set in a real Hausdorff locally convex topological vector space equals the closure of the convex hull of its extreme points, then diagrams the separating-functional and exposed-face contradiction.", "byteSize": 2275375, "caption": "Compact convex sets in real Hausdorff locally convex spaces are the closed convex hulls of their extreme points.", "description": "The central body uses a continuous extreme-boundary glow and dense chord laminations rather than a finite generator set. Four panels follow the source route: the easy inclusion, a hypothetical point outside the closed hull, continuous-linear-functional separation, and an exposed maximizing face whose extreme point already belongs to the closed hull.", "derivatives": [ { "byteSize": 99686, "height": 960, "mediaType": "image/webp", "publicPath": "/assets/library-theorems/krein-milman-theorem/theorem-poster-v1-640.webp", "sha256": "sha256:10ea18f66bb05665d98cdabad20f3ace3ca66c8adc39134e8bb35f8e9c380d9a", "width": 640 }, { "byteSize": 213136, "height": 1536, "mediaType": "image/webp", "publicPath": "/assets/library-theorems/krein-milman-theorem/theorem-poster-v1-1024.webp", "sha256": "sha256:c7620008a7346363f281e66b727cb473b1bdf88489c42457e134196fc98d81d9", "width": 1024 } ], "generatedAt": "2026-07-25T20:30:40Z", "height": 1536, "mediaType": "image/png", "modelRequested": "gpt-image-2", "publicPath": "/assets/library-theorems/krein-milman-theorem/theorem-poster-v1.png", "sha256": "sha256:f1dbe060951049f7b7e3d795e8f69028557b30c2d73a747e5d219510df7be6e9", "title": "Krein–Milman Theorem at a glance", "width": 1024, "kind": "theorem_poster", "transcript": "CONVEX ANALYSIS · LOCALLY CONVEX SPACES\nKREIN–MILMAN THEOREM\ns compact · s convex\nclosure(convexHull ℝ (extremePoints ℝ s)) = s\nEXTREME POINTS GENERATE THE CLOSED CONVEX SET\nHOW THE CHECKED ROUTE MOVES\n1 · The closed convex hull already lies in s\n2 · Suppose x in s lies outside that closed hull\n3 · Separate x by a continuous linear functional\n4 · A maximizing exposed face yields an extreme point\nCONTRADICTION · THAT POINT ALREADY LIES IN THE CLOSED HULL\nEXACT SCOPE\nReal Hausdorff locally convex topological vector space.\nCompact convex set; closure is part of the conclusion.\nNo finite-dimensional or finite-representation claim." }, "visual": { "alt": "A smooth asymmetric convex body has an entire luminous granular boundary and dense overlapping chords filling its interior, while a thin outer halo marks the closure.", "byteSize": 2766646, "caption": "The closure of all convex combinations of the extreme points recovers the compact convex set.", "description": "A continuous non-exhaustive boundary glow avoids implying finitely many generators. Dense chord laminations converge through the full body, and a thin closure halo preserves the exact theorem boundary without suggesting that closure can be dropped.", "derivatives": [ { "byteSize": 40250, "height": 427, "mediaType": "image/webp", "publicPath": "/assets/library-theorems/krein-milman-theorem/theorem-schematic-v1-640.webp", "sha256": "sha256:e3f7c2d12b10d3fb00891749dad42c8f7c01275f04014dc72dca1c2e8e0b314b", "width": 640 }, { "byteSize": 148462, "height": 800, "mediaType": "image/webp", "publicPath": "/assets/library-theorems/krein-milman-theorem/theorem-schematic-v1-1200.webp", "sha256": "sha256:1989c25abb86cb8047d37167ff95b7a61cdb45c1626e76fb5d51f3edf5144fdf", "width": 1200 } ], "generatedAt": "2026-07-25T20:30:40Z", "height": 1024, "mediaType": "image/png", "modelRequested": "gpt-image-2", "publicPath": "/assets/library-theorems/krein-milman-theorem/theorem-schematic-v1.png", "sha256": "sha256:8cbb9a22405a762965c3031d3ac1e8d06e35f329b46c7b75b31a99dcbd4313f2", "title": "The closed convex body generated by its extreme boundary", "width": 1536 }, "walkthrough": null, "whyLandmark": "Krein–Milman is a cornerstone of infinite-dimensional convexity: it says that compact convex structure is controlled by extreme points, while the necessary closure records the genuinely topological content. Mathlib's proof exposes the central Hahn–Banach separation and extreme-face architecture." }, "nonClaims": [ "The closure in closure (convexHull ℝ (s.extremePoints ℝ)) is part of the theorem and is not removed.", "The declaration does not assume finite dimension or prove a finite-dimensional strengthening.", "It does not assert that the set of extreme points is finite or that every point has a finite or uniformly bounded extreme-point representation.", "It does not extend the equality to arbitrary noncompact convex sets.", "ProofAtlas did not originate the theorem or Mathlib's declaration, and the generated explanation and visuals are not proof evidence." ], "publicExportEligible": true, "relationship": "upstream_direct", "schemaVersion": "library-theorem-entry-data.v1", "statementId": "statement.library.mathlib.krein-milman-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 compact convex set in a real Hausdorff locally convex topological vector space is the closure of the convex hull of its extreme points.", "targetId": "target.library.mathlib.krein-milman-theorem.v001", "title": "Krein–Milman Theorem", "upstreamOrigin": { "declarationName": "closure_convexHull_extremePoints", "immutableSourceUrl": "https://github.com/leanprover-community/mathlib4/blob/5e932f97dd25535344f80f9dd8da3aab83df0fe6/Mathlib/Analysis/Convex/KreinMilman.lean#L96", "packageName": "mathlib", "repositoryUrl": "https://github.com/leanprover-community/mathlib4", "sourceArtifactHash": "sha256:76a30354400127df51e995389646830f6960ae9c1bb33b2b91a5ad9ab5091e80", "sourceByteLength": 6493, "sourceCommit": "5e932f97dd25535344f80f9dd8da3aab83df0fe6", "sourceFile": "Mathlib/Analysis/Convex/KreinMilman.lean", "sourceLine": 96, "verificationKind": "git_worktree" }, "publicPresentationReview": { "status": "reviewed" } }