Existing in Mathlib · Geometry

Carathéodory's Convex Hull Theorem

Every point in the convex hull of a set already lies in the convex hull of some finite affine-independent subset of that set. Taken over all such finite supports, those smaller hulls fill the entire original convex hull.

convex hulls
finite supports
affine independence
minimum-cardinality supports
affine dependence
convex-hull monotonicity

Exact theorem

Exact Mathlib statement

theorem convexHull_eq_union {𝕜 : Type*} {E : Type u} [Field 𝕜] [LinearOrder 𝕜] [IsStrictOrderedRing 𝕜] [AddCommGroup E] [Module 𝕜 E] {s : Set E} : convexHull 𝕜 s = ⋃ (t : Finset E) (_ : ↑t ⊆ s) (_ : AffineIndependent 𝕜 ((↑) : t → E)), convexHull 𝕜 ↑t

The theorem at a glance

Carathéodory's convex hull theorem at a glance

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

An ivory convex-geometry poster places the exact finite affine-independent union statement above a convex envelope tiled by overlapping finite hull cells, then isolates one crossed-out redundant support point above a three-vertex support that still encloses the same interior point. — The convex hull of a set is the union of the convex hulls of its finite affine-independent subsets.

Statement structure

Statement and scope

Read the exact Mathlib declaration

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

Scientific figure illustrating: Projected finite-support hulls of varied shapes fill a convex envelope while one gold point remains inside a green support after a redundant point fades. — Varied finite-support hulls cover the convex envelope, and the highlighted target stays enclosed when one dependent support point is removed.

Why it matters

A mathematical landmark

Carathéodory's theorem is a central bridge between convexity and finite affine structure. Mathlib's selected endpoint exposes the conceptual heart directly: minimum-cardinality finite supports must be affine-independent, and their convex hulls collectively recover the entire hull.

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 Carathéodory convexity endpoint in a module over a linearly ordered field with the displayed strict ordered-ring hypotheses: convexHull 𝕜 s is the union of convexHull 𝕜 ↑t over finite t ⊆ s whose inclusion t → E is affine-independent. Equivalently, every point in the convex hull has some finite affine-independent support drawn from s. The selected declaration has no explicit dimension or cardinality bound, and it does not choose a unique support, provide barycentric coordinates, or give a constructive algorithm.

The selected declaration contains no explicit dimension-dependent or cardinality bound on the finite support.
It does not assert that the supporting finite subset is unique, canonical, or chosen uniformly for all points.
It does not return barycentric coordinates or a constructive algorithm for finding a support.
The planar visual projection does not add a two-dimensional or finite-dimensional hypothesis to the formal statement.
ProofAtlas is indexing an existing Mathlib theorem, and generated explanations and visuals are not proof evidence.

Source and local evidence

Where the theorem comes from

Existing declaration: convexHull_eq_union in mathlib
Relationship: The checked artifact is the upstream declaration itself
Local Lean evidence: artifact.library.mathlib.caratheodory-convex-hull-theorem.v001
Source: Open the pinned upstream reference