Convex cones in inner product spaces

We define Set.innerDualCone to be the cone consisting of all points y such that for all points x in a given set 0 ≤ ⟪ x, y ⟫.

Main statements

We prove the following theorems:

• ConvexCone.innerDualCone_of_innerDualCone_eq_self: The innerDualCone of the innerDualCone of a nonempty, closed, convex cone is itself.
• ConvexCone.hyperplane_separation_of_nonempty_of_isClosed_of_nmem: This variant of the hyperplane separation theorem states that given a nonempty, closed, convex cone K in a complete, real inner product space H and a point b disjoint from it, there is a vector y which separates b from K in the sense that for all points x in K, 0 ≤ ⟪x, y⟫_ℝ and ⟪y, b⟫_ℝ < 0. This is also a geometric interpretation of the Farkas lemma.

The dual cone

def Set.innerDualCone {H : Type u_5} [NormedAddCommGroup H] [InnerProductSpace ℝ H] (s : Set H) : ConvexCone ℝ H

The dual cone is the cone consisting of all points y such that for all points x in a given set 0 ≤ ⟪ x, y ⟫.

Equations

• Set.innerDualCone s = { carrier := {y : H | ∀ x ∈ s, 0 ≤ ⟪x, y⟫_ℝ}, smul_mem' := ⋯, add_mem' := ⋯ }

@[simp]

theorem mem_innerDualCone {H : Type u_5} [NormedAddCommGroup H] [InnerProductSpace ℝ H] (y : H) (s : Set H) : y ∈ Set.innerDualCone s ↔ ∀ x ∈ s, 0 ≤ ⟪x, y⟫_ℝ

@[simp]

theorem innerDualCone_empty {H : Type u_5} [NormedAddCommGroup H] [InnerProductSpace ℝ H] : Set.innerDualCone ∅ = ⊤

@[simp]

theorem innerDualCone_zero {H : Type u_5} [NormedAddCommGroup H] [InnerProductSpace ℝ H] : Set.innerDualCone 0 = ⊤

Dual cone of the convex cone {0} is the total space.

@[simp]

theorem innerDualCone_univ {H : Type u_5} [NormedAddCommGroup H] [InnerProductSpace ℝ H] : Set.innerDualCone Set.univ = 0

Dual cone of the total space is the convex cone {0}.

theorem innerDualCone_le_innerDualCone {H : Type u_5} [NormedAddCommGroup H] [InnerProductSpace ℝ H] (s : Set H) (t : Set H) (h : t ⊆ s) : Set.innerDualCone s ≤ Set.innerDualCone t

theorem pointed_innerDualCone {H : Type u_5} [NormedAddCommGroup H] [InnerProductSpace ℝ H] (s : Set H) : ConvexCone.Pointed (Set.innerDualCone s)

theorem innerDualCone_singleton {H : Type u_5} [NormedAddCommGroup H] [InnerProductSpace ℝ H] (x : H) : Set.innerDualCone {x} = ConvexCone.comap ((innerₛₗ ℝ) x) (ConvexCone.positive ℝ ℝ)

The inner dual cone of a singleton is given by the preimage of the positive cone under the linear map fun y ↦ ⟪x, y⟫.

theorem innerDualCone_union {H : Type u_5} [NormedAddCommGroup H] [InnerProductSpace ℝ H] (s : Set H) (t : Set H) : Set.innerDualCone (s ∪ t) = Set.innerDualCone s ⊓ Set.innerDualCone t

theorem innerDualCone_insert {H : Type u_5} [NormedAddCommGroup H] [InnerProductSpace ℝ H] (x : H) (s : Set H) : Set.innerDualCone (insert x s) = Set.innerDualCone {x} ⊓ Set.innerDualCone s

theorem innerDualCone_iUnion {H : Type u_5} [NormedAddCommGroup H] [InnerProductSpace ℝ H] {ι : Sort u_6} (f : ι → Set H) : Set.innerDualCone (⋃ (i : ι), f i) = ⨅ (i : ι), Set.innerDualCone (f i)

theorem innerDualCone_sUnion {H : Type u_5} [NormedAddCommGroup H] [InnerProductSpace ℝ H] (S : Set (Set H)) : Set.innerDualCone (⋃₀ S) = sInf (Set.innerDualCone '' S)

theorem innerDualCone_eq_iInter_innerDualCone_singleton {H : Type u_5} [NormedAddCommGroup H] [InnerProductSpace ℝ H] (s : Set H) : ↑(Set.innerDualCone s) = ⋂ (i : ↑s), ↑(Set.innerDualCone {↑i})

The dual cone of s equals the intersection of dual cones of the points in s.

theorem isClosed_innerDualCone {H : Type u_5} [NormedAddCommGroup H] [InnerProductSpace ℝ H] (s : Set H) : IsClosed ↑(Set.innerDualCone s)

theorem ConvexCone.pointed_of_nonempty_of_isClosed {H : Type u_5} [NormedAddCommGroup H] [InnerProductSpace ℝ H] (K : ConvexCone ℝ H) (ne : Set.Nonempty ↑K) (hc : IsClosed ↑K) : ConvexCone.Pointed K

theorem ConvexCone.hyperplane_separation_of_nonempty_of_isClosed_of_nmem {H : Type u_5} [NormedAddCommGroup H] [InnerProductSpace ℝ H] [CompleteSpace H] (K : ConvexCone ℝ H) (ne : Set.Nonempty ↑K) (hc : IsClosed ↑K) {b : H} (disj : b ∉ K) : ∃ (y : H), (∀ x ∈ K, 0 ≤ ⟪x, y⟫_ℝ) ∧ ⟪y, b⟫_ℝ < 0

This is a stronger version of the Hahn-Banach separation theorem for closed convex cones. This is also the geometric interpretation of Farkas' lemma.

theorem ConvexCone.innerDualCone_of_innerDualCone_eq_self {H : Type u_5} [NormedAddCommGroup H] [InnerProductSpace ℝ H] [CompleteSpace H] (K : ConvexCone ℝ H) (ne : Set.Nonempty ↑K) (hc : IsClosed ↑K) : Set.innerDualCone ↑(Set.innerDualCone ↑K) = K

The inner dual of inner dual of a non-empty, closed convex cone is itself.