mathlib documentation

analysis.convex.exposed

Exposed sets #

This file defines exposed sets and exposed points for sets in a real vector space.

An exposed subset of A is a subset of A that is the set of all maximal points of a functional (a continuous linear map E → 𝕜) over A. By convention, ∅ is an exposed subset of all sets. This allows for better functoriality of the definition (the intersection of two exposed subsets is exposed, faces of a polytope form a bounded lattice). This is an analytic notion of "being on the side of". It is stronger than being extreme (see is_exposed.is_extreme), but weaker (for exposed points) than being a vertex.

An exposed set of A is sometimes called a "face of A", but we decided to reserve this terminology to the more specific notion of a face of a polytope (sometimes hopefully soon out on mathlib!).

Main declarations #

is_exposed 𝕜 A B: States that B is an exposed set of A (in the literature, A is often implicit).
is_exposed.is_extreme: An exposed set is also extreme.

References #

See chapter 8 of Barry Simon, Convexity

TODO #

Define intrinsic frontier/interior and prove the lemmas related to exposed sets and points.

Generalise to Locally Convex Topological Vector Spaces™

More not-yet-PRed stuff is available on the branch sperner_again.

def is_exposed (𝕜 : Type u_1) {E : Type u_2} [normed_linear_ordered_field 𝕜] [normed_add_comm_group E] [normed_space 𝕜 E] (A B : set E) :

Prop

A set B is exposed with respect to A iff it maximizes some functional over A (and contains all points maximizing it). Written is_exposed 𝕜 A B.

Equations

is_exposed 𝕜 A B = (B.nonempty → (∃ (l : E →L[𝕜] 𝕜), B = {x ∈ A | ∀ (y : E), y ∈ A → ⇑l y ≤ ⇑l x}))

def continuous_linear_map.to_exposed {𝕜 : Type u_1} {E : Type u_2} [normed_linear_ordered_field 𝕜] [normed_add_comm_group E] [normed_space 𝕜 E] (l : E →L[𝕜] 𝕜) (A : set E) :

A useful way to build exposed sets from intersecting A with halfspaces (modelled by an inequality with a functional).

Equations

l.to_exposed A = {x ∈ A | ∀ (y : E), y ∈ A → ⇑l y ≤ ⇑l x}

theorem continuous_linear_map.to_exposed.is_exposed {𝕜 : Type u_1} {E : Type u_2} [normed_linear_ordered_field 𝕜] [normed_add_comm_group E] [normed_space 𝕜 E] {l : E →L[𝕜] 𝕜} {A : set E} :

is_exposed 𝕜 A (l.to_exposed A)

theorem is_exposed_empty {𝕜 : Type u_1} {E : Type u_2} [normed_linear_ordered_field 𝕜] [normed_add_comm_group E] [normed_space 𝕜 E] {A : set E} :

is_exposed 𝕜 A ∅

@[protected]

theorem is_exposed.subset {𝕜 : Type u_1} {E : Type u_2} [normed_linear_ordered_field 𝕜] [normed_add_comm_group E] [normed_space 𝕜 E] {A B : set E} (hAB : is_exposed 𝕜 A B) :

B ⊆ A

@[protected, refl]

theorem is_exposed.refl {𝕜 : Type u_1} {E : Type u_2} [normed_linear_ordered_field 𝕜] [normed_add_comm_group E] [normed_space 𝕜 E] (A : set E) :

is_exposed 𝕜 A A

@[protected]

theorem is_exposed.antisymm {𝕜 : Type u_1} {E : Type u_2} [normed_linear_ordered_field 𝕜] [normed_add_comm_group E] [normed_space 𝕜 E] {A B : set E} (hB : is_exposed 𝕜 A B) (hA : is_exposed 𝕜 B A) :

A = B

@[protected]

theorem is_exposed.mono {𝕜 : Type u_1} {E : Type u_2} [normed_linear_ordered_field 𝕜] [normed_add_comm_group E] [normed_space 𝕜 E] {A B C : set E} (hC : is_exposed 𝕜 A C) (hBA : B ⊆ A) (hCB : C ⊆ B) :

is_exposed 𝕜 B C

theorem is_exposed.eq_inter_halfspace {𝕜 : Type u_1} {E : Type u_2} [normed_linear_ordered_field 𝕜] [normed_add_comm_group E] [normed_space 𝕜 E] {A B : set E} (hAB : is_exposed 𝕜 A B) :

∃ (l : E →L[𝕜] 𝕜) (a : 𝕜), B = {x ∈ A | a ≤ ⇑l x}

If B is an exposed subset of A, then B is the intersection of A with some closed halfspace. The converse is not true. It would require that the corresponding open halfspace doesn't intersect A.

@[protected]

theorem is_exposed.inter {𝕜 : Type u_1} {E : Type u_2} [normed_linear_ordered_field 𝕜] [normed_add_comm_group E] [normed_space 𝕜 E] {A B C : set E} (hB : is_exposed 𝕜 A B) (hC : is_exposed 𝕜 A C) :

is_exposed 𝕜 A (B ∩ C)

theorem is_exposed.sInter {𝕜 : Type u_1} {E : Type u_2} [normed_linear_ordered_field 𝕜] [normed_add_comm_group E] [normed_space 𝕜 E] {A : set E} {F : finset (set E)} (hF : F.nonempty) (hAF : ∀ (B : set E), B ∈ F → is_exposed 𝕜 A B) :

is_exposed 𝕜 A (⋂₀ ↑F)

theorem is_exposed.inter_left {𝕜 : Type u_1} {E : Type u_2} [normed_linear_ordered_field 𝕜] [normed_add_comm_group E] [normed_space 𝕜 E] {A B C : set E} (hC : is_exposed 𝕜 A C) (hCB : C ⊆ B) :

is_exposed 𝕜 (A ∩ B) C

theorem is_exposed.inter_right {𝕜 : Type u_1} {E : Type u_2} [normed_linear_ordered_field 𝕜] [normed_add_comm_group E] [normed_space 𝕜 E] {A B C : set E} (hC : is_exposed 𝕜 B C) (hCA : C ⊆ A) :

is_exposed 𝕜 (A ∩ B) C

@[protected]

theorem is_exposed.is_extreme {𝕜 : Type u_1} {E : Type u_2} [normed_linear_ordered_field 𝕜] [normed_add_comm_group E] [normed_space 𝕜 E] {A B : set E} (hAB : is_exposed 𝕜 A B) :

is_extreme 𝕜 A B

@[protected]

theorem is_exposed.convex {𝕜 : Type u_1} {E : Type u_2} [normed_linear_ordered_field 𝕜] [normed_add_comm_group E] [normed_space 𝕜 E] {A B : set E} (hAB : is_exposed 𝕜 A B) (hA : convex 𝕜 A) :

convex 𝕜 B

@[protected]

theorem is_exposed.is_closed {𝕜 : Type u_1} {E : Type u_2} [normed_linear_ordered_field 𝕜] [normed_add_comm_group E] [normed_space 𝕜 E] {A B : set E} [order_closed_topology 𝕜] (hAB : is_exposed 𝕜 A B) (hA : is_closed A) :

@[protected]

theorem is_exposed.is_compact {𝕜 : Type u_1} {E : Type u_2} [normed_linear_ordered_field 𝕜] [normed_add_comm_group E] [normed_space 𝕜 E] {A B : set E} [order_closed_topology 𝕜] (hAB : is_exposed 𝕜 A B) (hA : is_compact A) :

def set.exposed_points (𝕜 : Type u_1) {E : Type u_2} [normed_linear_ordered_field 𝕜] [normed_add_comm_group E] [normed_space 𝕜 E] (A : set E) :

A point is exposed with respect to A iff there exists an hyperplane whose intersection with A is exactly that point.

Equations

set.exposed_points 𝕜 A = {x ∈ A | ∃ (l : E →L[𝕜] 𝕜), ∀ (y : E), y ∈ A → ⇑l y ≤ ⇑l x ∧ (⇑l x ≤ ⇑l y → y = x)}

theorem exposed_point_def {𝕜 : Type u_1} {E : Type u_2} [normed_linear_ordered_field 𝕜] [normed_add_comm_group E] [normed_space 𝕜 E] {A : set E} {x : E} :

x ∈ set.exposed_points 𝕜 A ↔ x ∈ A ∧ ∃ (l : E →L[𝕜] 𝕜), ∀ (y : E), y ∈ A → ⇑l y ≤ ⇑l x ∧ (⇑l x ≤ ⇑l y → y = x)

theorem exposed_points_subset {𝕜 : Type u_1} {E : Type u_2} [normed_linear_ordered_field 𝕜] [normed_add_comm_group E] [normed_space 𝕜 E] {A : set E} :

set.exposed_points 𝕜 A ⊆ A

@[simp]

theorem exposed_points_empty {𝕜 : Type u_1} {E : Type u_2} [normed_linear_ordered_field 𝕜] [normed_add_comm_group E] [normed_space 𝕜 E] :

set.exposed_points 𝕜 ∅ = ∅

theorem mem_exposed_points_iff_exposed_singleton {𝕜 : Type u_1} {E : Type u_2} [normed_linear_ordered_field 𝕜] [normed_add_comm_group E] [normed_space 𝕜 E] {A : set E} {x : E} :

x ∈ set.exposed_points 𝕜 A ↔ is_exposed 𝕜 A {x}

Exposed points exactly correspond to exposed singletons.

theorem exposed_points_subset_extreme_points {𝕜 : Type u_1} {E : Type u_2} [normed_linear_ordered_field 𝕜] [normed_add_comm_group E] [normed_space 𝕜 E] {A : set E} :

set.exposed_points 𝕜 A ⊆ set.extreme_points 𝕜 A