mathlib documentation

analysis.normed_space.hahn_banach.separation

Separation Hahn-Banach theorem #

In this file we prove the geometric Hahn-Banach theorem. For any two disjoint convex sets, there exists a continuous linear functional separating them, geometrically meaning that we can intercalate a plane between them.

We provide many variations to stricten the result under more assumptions on the convex sets:

TODO #

theorem separate_convex_open_set {E : Type u_2} [seminormed_add_comm_group E] [normed_space E] {s : set E} (hs₀ : 0 s) (hs₁ : convex s) (hs₂ : is_open s) {x₀ : E} (hx₀ : x₀ s) :
∃ (f : E →L[] ), f x₀ = 1 ∀ (x : E), x sf x < 1

Given a set s which is a convex neighbourhood of 0 and a point x₀ outside of it, there is a continuous linear functional f separating x₀ and s, in the sense that it sends x₀ to 1 and all of s to values strictly below 1.

theorem geometric_hahn_banach_open {E : Type u_2} [normed_add_comm_group E] [normed_space E] {s t : set E} (hs₁ : convex s) (hs₂ : is_open s) (ht : convex t) (disj : disjoint s t) :
∃ (f : E →L[] ) (u : ), (∀ (a : E), a sf a < u) ∀ (b : E), b tu f b

A version of the Hahn-Banach theorem: given disjoint convex sets s, t where s is open, there is a continuous linear functional which separates them.

theorem geometric_hahn_banach_open_point {E : Type u_2} [normed_add_comm_group E] [normed_space E] {s : set E} {x : E} (hs₁ : convex s) (hs₂ : is_open s) (disj : x s) :
∃ (f : E →L[] ), ∀ (a : E), a sf a < f x
theorem geometric_hahn_banach_point_open {E : Type u_2} [normed_add_comm_group E] [normed_space E] {t : set E} {x : E} (ht₁ : convex t) (ht₂ : is_open t) (disj : x t) :
∃ (f : E →L[] ), ∀ (b : E), b tf x < f b
theorem geometric_hahn_banach_open_open {E : Type u_2} [normed_add_comm_group E] [normed_space E] {s t : set E} (hs₁ : convex s) (hs₂ : is_open s) (ht₁ : convex t) (ht₃ : is_open t) (disj : disjoint s t) :
∃ (f : E →L[] ) (u : ), (∀ (a : E), a sf a < u) ∀ (b : E), b tu < f b
theorem geometric_hahn_banach_compact_closed {E : Type u_2} [normed_add_comm_group E] [normed_space E] {s t : set E} (hs₁ : convex s) (hs₂ : is_compact s) (ht₁ : convex t) (ht₂ : is_closed t) (disj : disjoint s t) :
∃ (f : E →L[] ) (u v : ), (∀ (a : E), a sf a < u) u < v ∀ (b : E), b tv < f b

A version of the Hahn-Banach theorem: given disjoint convex sets s, t where s is compact and t is closed, there is a continuous linear functional which strongly separates them.

theorem geometric_hahn_banach_closed_compact {E : Type u_2} [normed_add_comm_group E] [normed_space E] {s t : set E} (hs₁ : convex s) (hs₂ : is_closed s) (ht₁ : convex t) (ht₂ : is_compact t) (disj : disjoint s t) :
∃ (f : E →L[] ) (u v : ), (∀ (a : E), a sf a < u) u < v ∀ (b : E), b tv < f b

A version of the Hahn-Banach theorem: given disjoint convex sets s, t where s is closed, and t is compact, there is a continuous linear functional which strongly separates them.

theorem geometric_hahn_banach_point_closed {E : Type u_2} [normed_add_comm_group E] [normed_space E] {t : set E} {x : E} (ht₁ : convex t) (ht₂ : is_closed t) (disj : x t) :
∃ (f : E →L[] ) (u : ), f x < u ∀ (b : E), b tu < f b
theorem geometric_hahn_banach_closed_point {E : Type u_2} [normed_add_comm_group E] [normed_space E] {s : set E} {x : E} (hs₁ : convex s) (hs₂ : is_closed s) (disj : x s) :
∃ (f : E →L[] ) (u : ), (∀ (a : E), a sf a < u) u < f x
theorem geometric_hahn_banach_point_point {E : Type u_2} [normed_add_comm_group E] [normed_space E] {x y : E} (hxy : x y) :
∃ (f : E →L[] ), f x < f y

Special case of normed_space.eq_iff_forall_dual_eq.

theorem Inter_halfspaces_eq {E : Type u_2} [normed_add_comm_group E] [normed_space E] {s : set E} (hs₁ : convex s) (hs₂ : is_closed s) :
(⋂ (l : E →L[] ), {x : E | ∃ (y : E) (H : y s), l x l y}) = s

A closed convex set is the intersection of the halfspaces containing it.