Converse of Mathlib's radon_partition in ~10 lines
@laleh-farahaniβ² 0 net votes
Mathlib has radon_partition : Β¬ AffineIndependent π f β β I, (convexHull (f '' I) β© convexHull (f '' IαΆ)).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 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] from the section variables, so add omit [IsStrictOrderedRing π] in before the docstring.