# mathlib3documentation

analysis.convex.normed

# Topological and metric properties of convex sets in normed spaces #

THIS FILE IS SYNCHRONIZED WITH MATHLIB4. Any changes to this file require a corresponding PR to mathlib4.

We prove the following facts:

• convex_on_norm, convex_on_dist : norm and distance to a fixed point is convex on any convex set;
• convex_on_univ_norm, convex_on_univ_dist : norm and distance to a fixed point is convex on the whole space;
• 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.
theorem convex_on_norm {E : Type u_2} [ E] {s : set E} (hs : s) :

The norm on a real normed space is convex on any convex set. See also seminorm.convex_on and convex_on_univ_norm.

theorem convex_on_univ_norm {E : Type u_2} [ E] :

The norm on a real normed space is convex on the whole space. See also seminorm.convex_on and convex_on_norm.

theorem convex_on_dist {E : Type u_2} [ E] {s : set E} (z : E) (hs : s) :
(λ (z' : E), z)
theorem convex_on_univ_dist {E : Type u_2} [ E] (z : E) :
(λ (z' : E), z)
theorem convex_ball {E : Type u_2} [ E] (a : E) (r : ) :
r)
theorem convex_closed_ball {E : Type u_2} [ E] (a : E) (r : ) :
r)
theorem convex.thickening {E : Type u_2} [ E] {s : set E} (hs : s) (δ : ) :
s)
theorem convex.cthickening {E : Type u_2} [ E] {s : set E} (hs : s) (δ : ) :
s)
theorem convex_hull_exists_dist_ge {E : Type u_2} [ E] {s : set E} {x : E} (hx : x s) (y : E) :
(x' : E) (H : x' s), 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} [ E] {s t : set E} {x y : E} (hx : x s) (hy : y t) :
(x' : E) (H : x' s) (y' : E) (H : y' t), 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} [ 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} [ 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} [ E] {s : set E} :

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

@[protected, instance]
def normed_space.path_connected {E : Type u_2} [ E] :
@[protected, instance]
def normed_space.loc_path_connected {E : Type u_2} [ E] :
theorem dist_add_dist_of_mem_segment {E : Type u_2} [ E] {x y z : E} (h : y x z) :
+ =
theorem is_connected_set_of_same_ray {E : Type u_2} [ E] (x : E) :
is_connected {y : E | y}

The set of vectors in the same ray as x is connected.

theorem is_connected_set_of_same_ray_and_ne_zero {E : Type u_2} [ E] {x : E} (hx : x 0) :
is_connected {y : E | y y 0}

The set of nonzero vectors in the same ray as the nonzero vector x is connected.