Theorems · Definition · convex and discrete geometry

ProperCone.dual

{R : Type u_1} →
  {M : Type u_2} →
    {N : Type u_3} →
      [inst : CommRing R] →
        [inst_1 : PartialOrder R] →
          [inst_2 : IsOrderedRing R] →
            [inst_3 : TopologicalSpace R] →
              [ClosedIciTopology R] →
                [inst_5 : AddCommGroup M] →
                  [inst_6 : Module R M] →
                    [inst_7 : TopologicalSpace M] →
                      [inst_8 : AddCommGroup N] →
                        [inst_9 : Module R N] →
                          [inst_10 : TopologicalSpace N] →
                            (p : M →ₗ[R] N →ₗ[R] R) → [p.IsContPerfPair] → Set M → ProperCone R N

The dual cone of a set s with respect to a perfect pairing p is the cone consisting of all points y such that for all points x ∈ s we have 0 ≤ p x y.

Defined in

Mathlib.Analysis.Convex.Cone.Dual

Cited by

16 results in Mathlib

Foundations

Depth 76 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CommRingPartialOrderIsOrderedRingTopologicalSpaceClosedIciTopologyAddCommGroupModuleTopologicalSpaceAddCommGroupModuleTopologicalSpaceLinearMap.IsContPerfPair