mathlib documentation

analysis.convex.topology

Topological and metric properties of convex sets

We prove the following facts:

Standard simplex

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

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

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

Convex hull of a finite set is compact.

Convex hull of a finite set is closed.

Normed vector space

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

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

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

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

theorem convex_hull_exists_dist_ge2 {E : Type u_2} [normed_group E] [normed_space E] {s t : set E} {x y : E} :
x convex_hull sy convex_hull 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.

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

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

@[simp]

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] [normed_space E] {s : set E} :