File: Mathlib/Analysis/Convex/Normed.lean

full_name: convexOn_norm
code: /-- The norm on a real normed space is convex on any convex set. See also `Seminorm.convexOn`
and `convexOn_univ_norm`. -/
theorem convexOn_norm (hs : Convex ℝ s) : ConvexOn ℝ s norm 
goal: ConvexOn ℝ s norm

full_name: convexOn_univ_norm
code: /-- The norm on a real normed space is convex on the whole space. See also `Seminorm.convexOn`
and `convexOn_norm`. -/
theorem convexOn_univ_norm : ConvexOn ℝ univ (norm : E → ℝ) 
goal: ConvexOn ℝ Set.univ norm

full_name: convexOn_dist
code: theorem convexOn_dist (z : E) (hs : Convex ℝ s) : ConvexOn ℝ s fun z' => dist z' z 
goal: ConvexOn ℝ s fun z' => dist z' z

full_name: convexOn_univ_dist
code: theorem convexOn_univ_dist (z : E) : ConvexOn ℝ univ fun z' => dist z' z 
goal: ConvexOn ℝ Set.univ fun z' => dist z' z

full_name: convex_ball
code: theorem convex_ball (a : E) (r : ℝ) : Convex ℝ (Metric.ball a r) 
goal: Convex ℝ (Metric.ball a r)

full_name: convex_closedBall
code: theorem convex_closedBall (a : E) (r : ℝ) : Convex ℝ (Metric.closedBall a r) 
goal: Convex ℝ (Metric.closedBall a r)

full_name: Convex.thickening
code: theorem Convex.thickening (hs : Convex ℝ s) (δ : ℝ) : Convex ℝ (thickening δ s) 
goal: Convex ℝ (Metric.thickening δ s)

full_name: Convex.cthickening
code: theorem Convex.cthickening (hs : Convex ℝ s) (δ : ℝ) : Convex ℝ (cthickening δ s) 
goal: Convex ℝ (Metric.cthickening δ s)

full_name: convexHull_exists_dist_ge
code: /-- 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 convexHull_exists_dist_ge {s : Set E} {x : E} (hx : x ∈ convexHull ℝ s) (y : E) :
    ∃ x' ∈ s, dist x y ≤ dist x' y 
goal: ∃ x' ∈ s, dist x y ≤ dist x' y

full_name: convexHull_exists_dist_ge2
code: /-- 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`. -/
theorem convexHull_exists_dist_ge2 {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' 
goal: ∃ x' ∈ s, ∃ y' ∈ t, dist x y ≤ dist x' y'

full_name: convexHull_ediam
code: /-- Emetric diameter of the convex hull of a set `s` equals the emetric diameter of `s`. -/
@[simp]
theorem convexHull_ediam (s : Set E) : EMetric.diam (convexHull ℝ s) = EMetric.diam s 
goal: EMetric.diam ((convexHull ℝ) s) = EMetric.diam s

full_name: convexHull_diam
code: /-- Diameter of the convex hull of a set `s` equals the emetric diameter of `s`. -/
@[simp]
theorem convexHull_diam (s : Set E) : Metric.diam (convexHull ℝ s) = Metric.diam s 
goal: Metric.diam ((convexHull ℝ) s) = Metric.diam s

full_name: isBounded_convexHull
code: /-- Convex hull of `s` is bounded if and only if `s` is bounded. -/
@[simp]
theorem isBounded_convexHull {s : Set E} :
    Bornology.IsBounded (convexHull ℝ s) ↔ Bornology.IsBounded s 
goal: Bornology.IsBounded ((convexHull ℝ) s) ↔ Bornology.IsBounded s

full_name: NormedSpace.instPathConnectedSpace
code: instance (priority := 100) NormedSpace.instPathConnectedSpace : PathConnectedSpace E 
goal: PathConnectedSpace E

full_name: NormedSpace.instLocPathConnectedSpace
code: instance (priority := 100) NormedSpace.instLocPathConnectedSpace : LocPathConnectedSpace E 
goal: LocPathConnectedSpace E

full_name: Wbtw.dist_add_dist
code: theorem Wbtw.dist_add_dist {x y z : P} (h : Wbtw ℝ x y z) :
    dist x y + dist y z = dist x z 
goal: dist x y + dist y z = dist x z

full_name: dist_add_dist_of_mem_segment
code: theorem dist_add_dist_of_mem_segment {x y z : E} (h : y ∈ [x -[ℝ] z]) :
    dist x y + dist y z = dist x z 
goal: dist x y + dist y z = dist x z

full_name: isConnected_setOf_sameRay
code: /-- The set of vectors in the same ray as `x` is connected. -/
theorem isConnected_setOf_sameRay (x : E) : IsConnected { y | SameRay ℝ x y } 
goal: IsConnected {y | SameRay ℝ x y}

full_name: isConnected_setOf_sameRay_and_ne_zero
code: /-- The set of nonzero vectors in the same ray as the nonzero vector `x` is connected. -/
theorem isConnected_setOf_sameRay_and_ne_zero {x : E} (hx : x ≠ 0) :
    IsConnected { y | SameRay ℝ x y ∧ y ≠ 0 } 
goal: IsConnected {y | SameRay ℝ x y ∧ y ≠ 0}