Theorems · Theorem · convex and discrete geometry
ProperCone.relative_hyperplane_separation
∀ {E : Type u_2} {F : Type u_3} [inst : NormedAddCommGroup E] [inst_1 : InnerProductSpace ℝ E]
[inst_2 : CompleteSpace E] [inst_3 : NormedAddCommGroup F] [inst_4 : InnerProductSpace ℝ F] [inst_5 : CompleteSpace F]
{C : ProperCone ℝ E} {f : E →L[ℝ] F} {b : F},
b ∈ ProperCone.map f C ↔ ∀ (y : F), (ContinuousLinearMap.adjoint f) y ∈ ProperCone.innerDual ↑C → 0 ≤ inner ℝ b yRelative geometric interpretation of Farkas' lemma. Also stronger version of the Hahn-Banach separation theorem for proper cones.
- Defined in
- Mathlib.Analysis.Convex.Cone.InnerDual
- Cited by
- 1 results in Mathlib
- Foundations
- Depth 193 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites36
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
- Realstatement and proof · cited by 25,697
- RingHom.idstatement and proof · cited by 18,349
- NormedAddCommGroupstatement and proof · cited by 15,752
- SetLike.coestatement and proof · cited by 8,199
- Set.imageproof · cited by 5,609
- nhdsproof · cited by 5,554
- ContinuousLinearMapstatement and proof · cited by 5,352
- Filter.Tendstoproof · cited by 3,814
- InnerProductSpacestatement and proof · cited by 3,523
- CompleteSpacestatement and proof · cited by 2,532
- Filter.atTopproof · cited by 2,405
Cited by1
Results whose statement or proof uses this declaration.
- ProperCone.hyperplane_separation_of_notMemproof · cited by 0