Metric properties of convex sets in normed spaces

We prove the following facts:

• convexOn_norm, convexOn_dist : norm and distance to a fixed point is convex on any convex set;
• convexOn_univ_norm, convexOn_univ_dist : norm and distance to a fixed point is convex on the whole space;
• convexHull_ediam, convexHull_diam : convex hull of a set has the same (e)metric diameter as the original set;
• isBounded_convexHull : convex hull of a set is bounded if and only if the original set is bounded.

theorem convexOn_norm {E : Type u_1} [SeminormedAddCommGroup E] [NormedSpace ℝ E] {s : Set E} (hs : Convex ℝ s) : ConvexOn ℝ s norm

The norm on a real normed space is convex on any convex set. See also Seminorm.convexOn and convexOn_univ_norm.

theorem convexOn_univ_norm {E : Type u_1} [SeminormedAddCommGroup E] [NormedSpace ℝ E] : ConvexOn ℝ Set.univ norm

The norm on a real normed space is convex on the whole space. See also Seminorm.convexOn and convexOn_norm.

theorem convexOn_dist {E : Type u_1} [SeminormedAddCommGroup E] [NormedSpace ℝ E] {s : Set E} (z : E) (hs : Convex ℝ s) : ConvexOn ℝ s fun (z' : E) => dist z' z

theorem convexOn_univ_dist {E : Type u_1} [SeminormedAddCommGroup E] [NormedSpace ℝ E] (z : E) : ConvexOn ℝ Set.univ fun (z' : E) => dist z' z

theorem convex_ball {E : Type u_1} [SeminormedAddCommGroup E] [NormedSpace ℝ E] (a : E) (r : ℝ) : Convex ℝ (Metric.ball a r)

theorem convex_eball {E : Type u_1} [SeminormedAddCommGroup E] [NormedSpace ℝ E] (a : E) (r : ENNReal) : Convex ℝ (Metric.eball a r)

theorem convex_closedBall {E : Type u_1} [SeminormedAddCommGroup E] [NormedSpace ℝ E] (a : E) (r : ℝ) : Convex ℝ (Metric.closedBall a r)

theorem segment_subset_closedBall_left {E : Type u_1} [SeminormedAddCommGroup E] [NormedSpace ℝ E] (x y : E) : segment ℝ x y ⊆ Metric.closedBall x (dist x y)

The segment from x to y is contained in the closed ball centered at x with radius dist x y.

theorem segment_subset_closedBall_right {E : Type u_1} [SeminormedAddCommGroup E] [NormedSpace ℝ E] (x y : E) : segment ℝ x y ⊆ Metric.closedBall y (dist x y)

The segment from x to y is contained in the closed ball centered at y with radius dist x y.

theorem convex_closedEBall {E : Type u_1} [SeminormedAddCommGroup E] [NormedSpace ℝ E] (a : E) (r : ENNReal) : Convex ℝ (Metric.closedEBall a r)

theorem convexHull_sphere_eq_closedBall {F : Type u_2} [NormedAddCommGroup F] [NormedSpace ℝ F] [Nontrivial F] (x : F) {r : ℝ} (hr : 0 ≤ r) : (convexHull ℝ) (Metric.sphere x r) = Metric.closedBall x r

theorem convexHull_exists_dist_ge {E : Type u_1} [SeminormedAddCommGroup E] [NormedSpace ℝ E] {s : Set E} {x : E} (hx : x ∈ (convexHull ℝ) s) (y : E) : ∃ 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.thickening {E : Type u_1} [SeminormedAddCommGroup E] [NormedSpace ℝ E] {s : Set E} (hs : Convex ℝ s) (δ : ℝ) : Convex ℝ (Metric.thickening δ s)

theorem Convex.cthickening {E : Type u_1} [SeminormedAddCommGroup E] [NormedSpace ℝ E] {s : Set E} (hs : Convex ℝ s) (δ : ℝ) : Convex ℝ (Metric.cthickening δ s)

theorem convexHull_exists_dist_ge2 {E : Type u_1} [SeminormedAddCommGroup E] [NormedSpace ℝ E] {s t : Set E} {x y : E} (hx : x ∈ (convexHull ℝ) s) (hy : y ∈ (convexHull ℝ) t) : ∃ x' ∈ s, ∃ 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 convexHull_ediam {E : Type u_1} [SeminormedAddCommGroup E] [NormedSpace ℝ E] (s : Set E) : Metric.ediam ((convexHull ℝ) s) = Metric.ediam s

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

@[simp]

theorem convexHull_diam {E : Type u_1} [SeminormedAddCommGroup E] [NormedSpace ℝ E] (s : Set E) : Metric.diam ((convexHull ℝ) s) = Metric.diam s

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

@[simp]

theorem isBounded_convexHull {E : Type u_1} [SeminormedAddCommGroup E] [NormedSpace ℝ E] {s : Set E} : Bornology.IsBounded ((convexHull ℝ) s) ↔ Bornology.IsBounded s

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

@[instance 100]

instance NormedSpace.instPathConnectedSpace {E : Type u_1} [SeminormedAddCommGroup E] [NormedSpace ℝ E] : PathConnectedSpace E

theorem isConnected_setOf_sameRay {E : Type u_1} [SeminormedAddCommGroup E] [NormedSpace ℝ E] (x : E) : IsConnected {y : E | SameRay ℝ x y}

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

theorem isConnected_setOf_sameRay_and_ne_zero {E : Type u_1} [SeminormedAddCommGroup E] [NormedSpace ℝ E] {x : E} (hx : x ≠ 0) : IsConnected {y : E | SameRay ℝ x y ∧ y ≠ 0}

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

theorem norm_sub_le_of_mem_segment {E : Type u_1} [SeminormedAddCommGroup E] [NormedSpace ℝ E] {x y z : E} (hy : y ∈ segment ℝ x z) : ‖y - x‖ ≤ ‖z - x‖

theorem Filter.Eventually.segment_of_prod_nhds {E : Type u_1} [SeminormedAddCommGroup E] [NormedSpace ℝ E] {α : Type u_2} {f : Filter α} {x : E} {y z : α → E} {r : α → E → Prop} (hy : Tendsto y f (nhds x)) (hz : Tendsto z f (nhds x)) (hr : ∀ᶠ (p : α × E) in f ×ˢ nhds x, r p.1 p.2) : ∀ᶠ (χ : α) in f, ∀ v ∈ segment ℝ (y χ) (z χ), r χ v

theorem Filter.Eventually.segment_of_prod_nhdsWithin {E : Type u_1} [SeminormedAddCommGroup E] [NormedSpace ℝ E] {s : Set E} {α : Type u_2} {f : Filter α} {x : E} {y z : α → E} {r : α → E → Prop} (hy : Tendsto y f (nhds x)) (hz : Tendsto z f (nhds x)) (hr : ∀ᶠ (p : α × E) in f ×ˢ nhdsWithin x s, r p.1 p.2) (seg : ∀ᶠ (χ : α) in f, segment ℝ (y χ) (z χ) ⊆ s) : ∀ᶠ (χ : α) in f, ∀ v ∈ segment ℝ (y χ) (z χ), r χ v