Theorems · Theorem · convex and discrete geometry
convexIndependent_set_iff_notMem_convexHull_sdiff
∀ {𝕜 : Type u_1} {E : Type u_2} [inst : Semiring 𝕜] [inst_1 : PartialOrder 𝕜] [inst_2 : AddCommGroup E]
[inst_3 : Module 𝕜 E] {s : Set E}, ConvexIndependent 𝕜 Subtype.val ↔ ∀ x ∈ s, x ∉ (convexHull 𝕜) (s \ {x})If a set is convex independent, a point in the set is not in the convex hull of the other
points. See convexIndependent_iff_notMem_convexHull_sdiff for the indexed family version.
- Defined in
- Mathlib.Analysis.Convex.Independent
- Cited by
- 2 results in Mathlib
- Foundations
- Depth 63 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites14
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- DFunLike.coestatement and proof · cited by 62,936
- Setstatement and proof · cited by 53,352
- Modulestatement and proof · cited by 20,661
- Semiringstatement and proof · cited by 13,802
- AddCommGroupstatement and proof · cited by 12,871
- PartialOrderstatement and proof · cited by 6,410
- ClosureOperatorstatement · cited by 371
- Set.mem_singletonproof · cited by 183
- convexHullstatement and proof · cited by 163
- Set.sdiff_subsetproof · cited by 156
- ConvexIndependentstatement · cited by 15
- Set.subset_sdiff_singletonproof · cited by 14
Cited by2
Results whose statement or proof uses this declaration.
- convexIndependent_set_iff_notMem_convexHull_diffproof · cited by 0
- Convex.convexIndependent_extremePointsproof · cited by 0