# mathlibdocumentation

analysis.convex.extreme

# Extreme sets #

This file defines extreme sets and extreme points for sets in a module.

An extreme set of A is a subset of A that is as far as it can get in any outward direction: If point x is in it and point y β A, then the line passing through x and y leaves A at x. This is an analytic notion of "being on the side of". It is weaker than being exposed (see is_exposed.is_extreme).

## Main declarations #

• is_extreme π A B: States that B is an extreme set of A (in the literature, A is often implicit).
• set.extreme_points π A: Set of extreme points of A (corresponding to extreme singletons).
• convex.mem_extreme_points_iff_convex_diff: A useful equivalent condition to being an extreme point: x is an extreme point iff A \ {x} is convex.

## Implementation notes #

The exact definition of extremeness has been carefully chosen so as to make as many lemmas unconditional (in particular, the Krein-Milman theorem doesn't need the set to be convex!). In practice, A is often assumed to be a convex set.

## References #

See chapter 8 of Barry Simon, Convexity

## TODO #

Define intrinsic frontier and prove lemmas related to extreme sets and points.

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

def is_extreme (π : Type u_1) {E : Type u_2} [ordered_semiring π] [has_smul π E] (A B : set E) :
Prop

A set B is an extreme subset of A if B β A and all points of B only belong to open segments whose ends are in B.

Equations
Instances for is_extreme
def set.extreme_points (π : Type u_1) {E : Type u_2} [ordered_semiring π] [has_smul π E] (A : set E) :
set E

A point x is an extreme point of a set A if x belongs to no open segment with ends in A, except for the obvious open_segment x x.

Equations
• A = {x β A | β β¦xβ : Eβ¦, xβ β A β β β¦xβ : Eβ¦, xβ β A β x β open_segment π xβ xβ β xβ = x β§ xβ = x}
@[protected, refl]
theorem is_extreme.refl (π : Type u_1) {E : Type u_2} [ordered_semiring π] [has_smul π E] (A : set E) :
is_extreme π A A
@[protected]
theorem is_extreme.rfl {π : Type u_1} {E : Type u_2} [ordered_semiring π] [has_smul π E] {A : set E} :
is_extreme π A A
@[protected, trans]
theorem is_extreme.trans {π : Type u_1} {E : Type u_2} [ordered_semiring π] [has_smul π E] {A B C : set E} (hAB : is_extreme π A B) (hBC : is_extreme π B C) :
is_extreme π A C
@[protected]
theorem is_extreme.antisymm {π : Type u_1} {E : Type u_2} [ordered_semiring π] [has_smul π E] :
@[protected, instance]
def is_extreme.is_partial_order {π : Type u_1} {E : Type u_2} [ordered_semiring π] [has_smul π E] :
(is_extreme π)
theorem is_extreme.inter {π : Type u_1} {E : Type u_2} [ordered_semiring π] [has_smul π E] {A B C : set E} (hAB : is_extreme π A B) (hAC : is_extreme π A C) :
is_extreme π A (B β© C)
@[protected]
theorem is_extreme.mono {π : Type u_1} {E : Type u_2} [ordered_semiring π] [has_smul π E] {A B C : set E} (hAC : is_extreme π A C) (hBA : B β A) (hCB : C β B) :
is_extreme π B C
theorem is_extreme_Inter {π : Type u_1} {E : Type u_2} [ordered_semiring π] [has_smul π E] {A : set E} {ΞΉ : Type u_3} [nonempty ΞΉ] {F : ΞΉ β set E} (hAF : β (i : ΞΉ), is_extreme π A (F i)) :
is_extreme π A (β (i : ΞΉ), F i)
theorem is_extreme_bInter {π : Type u_1} {E : Type u_2} [ordered_semiring π] [has_smul π E] {A : set E} {F : set (set E)} (hF : F.nonempty) (hAF : β (B : set E), B β F β is_extreme π A B) :
is_extreme π A (β (B : set E) (H : B β F), B)
theorem is_extreme_sInter {π : Type u_1} {E : Type u_2} [ordered_semiring π] [has_smul π E] {A : set E} {F : set (set E)} (hF : F.nonempty) (hAF : β (B : set E), B β F β is_extreme π A B) :
is_extreme π A (ββ F)
theorem extreme_points_def {π : Type u_1} {E : Type u_2} [ordered_semiring π] [has_smul π E] {A : set E} {x : E} :
x β A β x β A β§ β (xβ : E), xβ β A β β (xβ : E), xβ β A β x β open_segment π xβ xβ β xβ = x β§ xβ = x
theorem mem_extreme_points_iff_extreme_singleton {π : Type u_1} {E : Type u_2} [ordered_semiring π] [has_smul π E] {A : set E} {x : E} :
x β A β is_extreme π A {x}

x is an extreme point to A iff {x} is an extreme set of A.

theorem extreme_points_subset {π : Type u_1} {E : Type u_2} [ordered_semiring π] [has_smul π E] {A : set E} :
A β A
@[simp]
theorem extreme_points_empty {π : Type u_1} {E : Type u_2} [ordered_semiring π] [has_smul π E] :
@[simp]
theorem extreme_points_singleton {π : Type u_1} {E : Type u_2} [ordered_semiring π] [has_smul π E] {x : E} :
{x} = {x}
theorem inter_extreme_points_subset_extreme_points_of_subset {π : Type u_1} {E : Type u_2} [ordered_semiring π] [has_smul π E] {A B : set E} (hBA : B β A) :
theorem is_extreme.extreme_points_subset_extreme_points {π : Type u_1} {E : Type u_2} [ordered_semiring π] [has_smul π E] {A B : set E} (hAB : is_extreme π A B) :
B β A
theorem is_extreme.extreme_points_eq {π : Type u_1} {E : Type u_2} [ordered_semiring π] [has_smul π E] {A B : set E} (hAB : is_extreme π A B) :
A useful restatement using segment: x is an extreme point iff the only (closed) segments that contain it are those with x as one of their endpoints.