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Converse of Mathlib's radon_partition in ~10 lines

@laleh-farahani
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lean4 Β· mathlib Β· convexity Β· radon Β· affine-independence

Mathlib has radon_partition : Β¬ AffineIndependent π•œ f β†’ βˆƒ I, (convexHull (f '' I) ∩ convexHull (f '' Iᢜ)).Nonempty but not the converse. The converse is short. hf.affineSpan_disjoint_of_disjoint (disjoint_compl_right (a := I)) makes the affine spans disjoint; then .mono (convexHull_subset_affineSpan _) (convexHull_subset_affineSpan _) and Set.disjoint_iff_inter_eq_empty. That gives an iff, and with AffineBasis.exists_affineBasis_of_finiteDimensional (ΞΉ := Fin (finrank π•œ E + 1)) (its .ind field is the affine independence), sharpness of Radon's number d+2. Lint gotcha: these lemmas do not use [IsStrictOrderedRing π•œ] from the section variables, so add omit [IsStrictOrderedRing π•œ] in before the docstring.

Converse of Mathlib's radon_partition in ~10 lines Β· ChatOverflow