# mathlibdocumentation

analysis.convex.topology

# 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;

### Standard simplex

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

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

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

std_simplex ι is bounded.

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

std_simplex ι is closed.

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

std_simplex ι is compact.

### Topological vector space

theorem convex.interior {E : Type u_2} [ E] {s : set E} :
convex (interior s)

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

theorem convex.closure {E : Type u_2} [ E] {s : set E} :
convex (closure s)

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

theorem set.finite.compact_convex_hull {E : Type u_2} [ E] {s : set E} :
s.finite

Convex hull of a finite set is compact.

theorem set.finite.is_closed_convex_hull {E : Type u_2} [ E] [t2_space E] {s : set E} :
s.finite

Convex hull of a finite set is closed.

### Normed vector space

theorem convex_on_dist {E : Type u_2} [normed_group E] [ E] (z : E) (s : set E) :
(λ (z' : E), dist z' z)

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

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

theorem convex_hull_exists_dist_ge {E : Type u_2} [normed_group E] [ E] {s : set E} {x : E} (hx : x ) (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.

theorem convex_hull_exists_dist_ge2 {E : Type u_2} [normed_group E] [ E] {s t : set E} {x y : E} :
x y (∃ (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.

@[simp]
theorem convex_hull_ediam {E : Type u_2} [normed_group E] [ E] (s : set E) :

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

@[simp]
theorem convex_hull_diam {E : Type u_2} [normed_group E] [ E] (s : set E) :

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

@[simp]
theorem bounded_convex_hull {E : Type u_2} [normed_group E] [ E] {s : set E} :

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

theorem convex.is_path_connected {E : Type u_2} [normed_group E] [ E] {s : set E} :

@[instance]
def normed_space.path_connected {E : Type u_2} [normed_group E] [ E] :

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