Topological and metric properties of convex sets #

We prove the following facts:

convex.interior : interior of a convex set is convex;
convex.closure : closure of a convex set is convex;
set.finite.compact_convex_hull : convex hull of a finite set is compact;
set.finite.is_closed_convex_hull : convex hull of a finite set is closed;
convex_on_dist : distance to a fixed point is convex on any convex set;
convex_hull_ediam, convex_hull_diam : convex hull of a set has the same (e)metric diameter as the original set;
bounded_convex_hull : convex hull of a set is bounded if and only if the original set is bounded.
bounded_std_simplex, is_closed_std_simplex, compact_std_simplex: topological properties of the standard simplex;

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theorem real.convex_iff_is_preconnected {s : set ℝ} :

convex ℝ s ↔ is_preconnected s

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theorem is_preconnected.convex {s : set ℝ} :

is_preconnected s → convex ℝ s

Alias of real.convex_iff_is_preconnected.

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theorem convex.is_preconnected {s : set ℝ} :

convex ℝ s → is_preconnected s

Alias of real.convex_iff_is_preconnected.

Standard simplex #

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theorem std_simplex_subset_closed_ball {ι : Type u_1} [fintype ι] :

std_simplex ℝ ι ⊆ metric.closed_ball 0 1

Every vector in std_simplex 𝕜 ι has max-norm at most 1.

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theorem bounded_std_simplex (ι : Type u_1) [fintype ι] :

metric.bounded (std_simplex ℝ ι)

std_simplex ℝ ι is bounded.

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theorem is_closed_std_simplex (ι : Type u_1) [fintype ι] :

is_closed (std_simplex ℝ ι)

std_simplex ℝ ι is closed.

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theorem compact_std_simplex (ι : Type u_1) [fintype ι] :

is_compact (std_simplex ℝ ι)

std_simplex ℝ ι is compact.

Topological vector space #

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theorem convex.combo_interior_self_subset_interior {E : Type u_2} [add_comm_group E] [module ℝ E] [topological_space E] [topological_add_group E] [has_continuous_smul ℝ E] {s : set E} (hs : convex ℝ s) {a b : ℝ} (ha : 0 < a) (hb : 0 ≤ b) (hab : a + b = 1) :

a • interior s + b • s ⊆ interior s

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theorem convex.combo_self_interior_subset_interior {E : Type u_2} [add_comm_group E] [module ℝ E] [topological_space E] [topological_add_group E] [has_continuous_smul ℝ E] {s : set E} (hs : convex ℝ s) {a b : ℝ} (ha : 0 ≤ a) (hb : 0 < b) (hab : a + b = 1) :

a • s + b • interior s ⊆ interior s

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theorem convex.combo_mem_interior_left {E : Type u_2} [add_comm_group E] [module ℝ E] [topological_space E] [topological_add_group E] [has_continuous_smul ℝ E] {s : set E} (hs : convex ℝ s) {x y : E} (hx : x ∈ interior s) (hy : y ∈ s) {a b : ℝ} (ha : 0 < a) (hb : 0 ≤ b) (hab : a + b = 1) :

a • x + b • y ∈ interior s

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theorem convex.combo_mem_interior_right {E : Type u_2} [add_comm_group E] [module ℝ E] [topological_space E] [topological_add_group E] [has_continuous_smul ℝ E] {s : set E} (hs : convex ℝ s) {x y : E} (hx : x ∈ s) (hy : y ∈ interior s) {a b : ℝ} (ha : 0 ≤ a) (hb : 0 < b) (hab : a + b = 1) :

a • x + b • y ∈ interior s

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theorem convex.open_segment_subset_interior_left {E : Type u_2} [add_comm_group E] [module ℝ E] [topological_space E] [topological_add_group E] [has_continuous_smul ℝ E] {s : set E} (hs : convex ℝ s) {x y : E} (hx : x ∈ interior s) (hy : y ∈ s) :

open_segment ℝ x y ⊆ interior s

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theorem convex.open_segment_subset_interior_right {E : Type u_2} [add_comm_group E] [module ℝ E] [topological_space E] [topological_add_group E] [has_continuous_smul ℝ E] {s : set E} (hs : convex ℝ s) {x y : E} (hx : x ∈ s) (hy : y ∈ interior s) :

open_segment ℝ x y ⊆ interior s

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theorem convex.add_smul_sub_mem_interior {E : Type u_2} [add_comm_group E] [module ℝ E] [topological_space E] [topological_add_group E] [has_continuous_smul ℝ E] {s : set E} (hs : convex ℝ s) {x y : E} (hx : x ∈ s) (hy : y ∈ interior s) {t : ℝ} (ht : t ∈ set.Ioc 0 1) :

x + t • (y - x) ∈ interior s

If x ∈ s and y ∈ interior s, then the segment (x, y] is included in interior s.

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theorem convex.add_smul_mem_interior {E : Type u_2} [add_comm_group E] [module ℝ E] [topological_space E] [topological_add_group E] [has_continuous_smul ℝ E] {s : set E} (hs : convex ℝ s) {x y : E} (hx : x ∈ s) (hy : x + y ∈ interior s) {t : ℝ} (ht : t ∈ set.Ioc 0 1) :

x + t • y ∈ interior s

If x ∈ s and x + y ∈ interior s, then x + t y ∈ interior s for t ∈ (0, 1].

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theorem convex.interior {E : Type u_2} [add_comm_group E] [module ℝ E] [topological_space E] [topological_add_group E] [has_continuous_smul ℝ E] {s : set E} (hs : convex ℝ s) :

convex ℝ (interior s)

In a topological vector space, the interior of a convex set is convex.

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theorem convex.closure {E : Type u_2} [add_comm_group E] [module ℝ E] [topological_space E] [topological_add_group E] [has_continuous_smul ℝ E] {s : set E} (hs : convex ℝ s) :

convex ℝ (closure s)

In a topological vector space, the closure of a convex set is convex.

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theorem set.finite.compact_convex_hull {E : Type u_2} [add_comm_group E] [module ℝ E] [topological_space E] [topological_add_group E] [has_continuous_smul ℝ E] {s : set E} (hs : s.finite) :

is_compact (⇑(convex_hull ℝ) s)

Convex hull of a finite set is compact.

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theorem set.finite.is_closed_convex_hull {E : Type u_2} [add_comm_group E] [module ℝ E] [topological_space E] [topological_add_group E] [has_continuous_smul ℝ E] [t2_space E] {s : set E} (hs : s.finite) :

is_closed (⇑(convex_hull ℝ) s)

Convex hull of a finite set is closed.

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theorem convex.subset_interior_image_homothety_of_one_lt {E : Type u_2} [add_comm_group E] [module ℝ E] [topological_space E] [topological_add_group E] [has_continuous_smul ℝ E] {s : set E} (hs : convex ℝ s) {x : E} (hx : x ∈ interior s) (t : ℝ) (ht : 1 < t) :

s ⊆ interior (⇑(affine_map.homothety x t) '' s)

If we dilate a convex set about a point in its interior by a scale t > 1, the interior of the result contains the original set.

TODO Generalise this from convex sets to sets that are balanced / star-shaped about x.

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theorem convex.is_path_connected {E : Type u_2} [add_comm_group E] [module ℝ E] [topological_space E] [topological_add_group E] [has_continuous_smul ℝ E] {s : set E} (hconv : convex ℝ s) (hne : s.nonempty) :

is_path_connected s

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@[protected, instance]

def topological_add_group.path_connected {E : Type u_2} [add_comm_group E] [module ℝ E] [topological_space E] [topological_add_group E] [has_continuous_smul ℝ E] :

path_connected_space E

Normed vector space #

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theorem convex_on_dist {E : Type u_2} [normed_group E] [normed_space ℝ E] (z : E) (s : set E) (hs : convex ℝ s) :

convex_on ℝ s (λ (z' : E), dist z' z)

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theorem convex_ball {E : Type u_2} [normed_group E] [normed_space ℝ E] (a : E) (r : ℝ) :

convex ℝ (metric.ball a r)

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theorem convex_closed_ball {E : Type u_2} [normed_group E] [normed_space ℝ E] (a : E) (r : ℝ) :

convex ℝ (metric.closed_ball a r)

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theorem convex_hull_exists_dist_ge {E : Type u_2} [normed_group E] [normed_space ℝ E] {s : set E} {x : E} (hx : x ∈ ⇑(convex_hull ℝ) s) (y : E) :

∃ (x' : E) (H : x' ∈ s), dist x y ≤ dist x' y

Given a point x in the convex hull of s and a point y, there exists a point of s at distance at least dist x y from y.

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theorem convex_hull_exists_dist_ge2 {E : Type u_2} [normed_group E] [normed_space ℝ E] {s t : set E} {x y : E} (hx : x ∈ ⇑(convex_hull ℝ) s) (hy : y ∈ ⇑(convex_hull ℝ) t) :

∃ (x' : E) (H : x' ∈ s) (y' : E) (H : y' ∈ t), dist x y ≤ dist x' y'

Given a point x in the convex hull of s and a point y in the convex hull of t, there exist points x' ∈ s and y' ∈ t at distance at least dist x y.

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@[simp]

theorem convex_hull_ediam {E : Type u_2} [normed_group E] [normed_space ℝ E] (s : set E) :

emetric.diam (⇑(convex_hull ℝ) s) = emetric.diam s

Emetric diameter of the convex hull of a set s equals the emetric diameter of `s.

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@[simp]

theorem convex_hull_diam {E : Type u_2} [normed_group E] [normed_space ℝ E] (s : set E) :

metric.diam (⇑(convex_hull ℝ) s) = metric.diam s

Diameter of the convex hull of a set s equals the emetric diameter of `s.

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@[simp]

theorem bounded_convex_hull {E : Type u_2} [normed_group E] [normed_space ℝ E] {s : set E} :

metric.bounded (⇑(convex_hull ℝ) s) ↔ metric.bounded s

Convex hull of s is bounded if and only if s is bounded.

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@[protected, instance]

def normed_space.loc_path_connected {E : Type u_2} [normed_group E] [normed_space ℝ E] :

loc_path_connected_space E