Public Mathlib landmark · existing upstream theorem · read-only
ProofAtlas

Geometry · existing Mathlib declaration

Radon's Theorem

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.

The theorem at a glance

Radon'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.

Affine dependence yields a complementary split of the index set whose two image convex hulls intersect. Generated explanation only; the exact HTML statement and pinned source are authoritative.
Detailed visual 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.

Formal orientation

Statement map

Statement map for Radon's TheoremEvery affinely dependent indexed family admits a complementary split whose two image convex hulls have a common point. The pinned upstream declaration is Convex.radon_partition. The exact checked statement is theorem Convex.radon_partition {ι 𝕜 E : Type*} [Field 𝕜] [LinearOrder 𝕜] [IsStrictOrderedRing 𝕜] [AddCommGroup E] [Module 𝕜 E] {f : ι → E} (h : ¬ AffineIndependent 𝕜 f) : ∃ I, (convexHull 𝕜 (f '' I) ∩ convexHull 𝕜 (f '' Iᶜ)).Nonempty.Mathematical readingEvery affinely dependentindexed family admits acomplementary splitwhose two image convexhulls have a commonpoint.Pinned declarationmathlib ·Convex.radon_partitionExact checked formtheoremConvex.radon_partition{ι 𝕜 E : Type*} [Field𝕜] [LinearOrder 𝕜][IsStrictOrderedRing 𝕜][AddCommGroup E] [Module𝕜 E] {f : ι → E} (h : ¬AffineIndependent 𝕜 f) :∃ I, (convexHull 𝕜 (f ''I) ∩ convexHull 𝕜 (f ''Iᶜ)).NonemptyStatement map for Radon's TheoremEvery affinely dependent indexed family admits a complementary split whose two image convex hulls have a common point. The pinned upstream declaration is Convex.radon_partition. The exact checked statement is theorem Convex.radon_partition {ι 𝕜 E : Type*} [Field 𝕜] [LinearOrder 𝕜] [IsStrictOrderedRing 𝕜] [AddCommGroup E] [Module 𝕜 E] {f : ι → E} (h : ¬ AffineIndependent 𝕜 f) : ∃ I, (convexHull 𝕜 (f '' I) ∩ convexHull 𝕜 (f '' Iᶜ)).Nonempty.Mathematical readingEvery affinely dependentindexed family admits acomplementary splitwhose two image convexhulls have a commonpoint.Pinned declarationmathlib ·Convex.radon_partitionExact checked formtheoremConvex.radon_partition{ι 𝕜 E : Type*} [Field𝕜] [LinearOrder 𝕜][IsStrictOrderedRing 𝕜][AddCommGroup E] [Module𝕜 E] {f : ι → E} (h : ¬AffineIndependent 𝕜 f) :∃ I, (convexHull 𝕜 (f ''I) ∩ convexHull 𝕜 (f ''Iᶜ)).Nonempty

This deterministic map orients the reader from the mathematical summary to the pinned declaration and exact checked form. The HTML statement below is authoritative.

Complementary convex hull segments meet at a common point. Scientific diagram · explanatory, not proof evidence.
Detailed visual description

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.

Complementary intuition

A mathematical landmark

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.

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.radon_partition {ι 𝕜 E : Type*} [Field 𝕜] [LinearOrder 𝕜] [IsStrictOrderedRing 𝕜] [AddCommGroup E] [Module 𝕜 E] {f : ι → E} (h : ¬ AffineIndependent 𝕜 f) : ∃ I, (convexHull 𝕜 (f '' I) ∩ convexHull 𝕜 (f '' Iᶜ)).Nonempty
Selected evidence declaration text
theorem Convex.radon_partition {ι 𝕜 E : Type*} [Field 𝕜] [LinearOrder 𝕜] [IsStrictOrderedRing 𝕜] [AddCommGroup E] [Module 𝕜 E] {f : ι → E} (h : ¬ AffineIndependent 𝕜 f) : ∃ I, (convexHull 𝕜 (f '' I) ∩ convexHull 𝕜 (f '' Iᶜ)).Nonempty

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

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 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.

Provenance

Origin and evidence stay separate

Existing declaration
Convex.radon_partition in mathlib
Relationship
The checked artifact is the upstream declaration itself
Preferred checked artifact
artifact.library.mathlib.radon-theorem.v001
Source handling
Verified upstream reference; no mirrored source package