-- Mathlib/Analysis/Normed/Affine/Convex.lean
/-- We can intercalate a simplex between a point and one of its neighborhoods. -/
lemma exists_mem_interior_convexHull_affineBasis (hs : s ∈ 𝓝 x) :
    ∃ b : AffineBasis (Fin (finrank ℝ E + 1)) ℝ E,
      x ∈ interior (convexHull ℝ (range b)) ∧ convexHull ℝ (range b) ⊆ s

Proof state:
case intro.intro.intro
E✝ : Type u_1
inst✝⁴ : NormedAddCommGroup E✝
inst✝³ : NormedSpace ℝ E✝
s✝ : Set E✝
x : E✝
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : FiniteDimensional ℝ E
s : Set E
hs : s ∈ 𝓝 0
b : AffineBasis (Fin (finrank ℝ E + 1)) ℝ E
c : AffineBasis (Fin (finrank ℝ E + 1)) ℝ E := -Finset.centroid ℝ Finset.univ ⇑b +ᵥ b
hc₀ : 0 ∈ interior ((convexHull ℝ) (range ⇑c))
cnorm : ℝ := Finset.univ.sup' ⋯ fun i => ‖c i‖
hcnorm : range ⇑c ⊆ closedBall 0 (cnorm + 1)
ε : ℝ
hε : ε > 0
hεs : ball 0 ε ⊆ s
ε' : ℝ := ε / 2 / (cnorm + 1)
⊢ ∃ b, 0 ∈ interior ((convexHull ℝ) (range ⇑b)) ∧ (convexHull ℝ) (range ⇑b) ⊆ s

Tactic:
have hc' : 0 < cnorm + 1 := by
  have : 0 ≤ cnorm := Finset.le_sup'_of_le _ (Finset.mem_univ 0) (norm_nonneg _)
  positivity

Result:
case intro.intro.intro
E✝ : Type u_1
inst✝⁴ : NormedAddCommGroup E✝
inst✝³ : NormedSpace ℝ E✝
s✝ : Set E✝
x : E✝
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : FiniteDimensional ℝ E
s : Set E
hs : s ∈ 𝓝 0
b : AffineBasis (Fin (finrank ℝ E + 1)) ℝ E
c : AffineBasis (Fin (finrank ℝ E + 1)) ℝ E := -Finset.centroid ℝ Finset.univ ⇑b +ᵥ b
hc₀ : 0 ∈ interior ((convexHull ℝ) (range ⇑c))
cnorm : ℝ := Finset.univ.sup' ⋯ fun i => ‖c i‖
hcnorm : range ⇑c ⊆ closedBall 0 (cnorm + 1)
ε : ℝ
hε : ε > 0
hεs : ball 0 ε ⊆ s
ε' : ℝ := ε / 2 / (cnorm + 1)
hc' : 0 < cnorm + 1
⊢ ∃ b, 0 ∈ interior ((convexHull ℝ) (range ⇑b)) ∧ (convexHull ℝ) (range ⇑b) ⊆ s

Proof state:
case intro
E✝ : Type u_1
inst✝⁴ : NormedAddCommGroup E✝
inst✝³ : NormedSpace ℝ E✝
s✝ : Set E✝
x : E✝
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : FiniteDimensional ℝ E
s : Set E
hs : s ∈ 𝓝 0
b : AffineBasis (Fin (finrank ℝ E + 1)) ℝ E
c : AffineBasis (Fin (finrank ℝ E + 1)) ℝ E := -Finset.centroid ℝ Finset.univ ⇑b +ᵥ b
hc₀ : 0 ∈ interior ((convexHull ℝ) (range ⇑c))
cnorm : ℝ := Finset.univ.sup' ⋯ fun i => ‖c i‖
hcnorm : range ⇑c ⊆ closedBall 0 (cnorm + 1)
⊢ ∃ b, 0 ∈ interior ((convexHull ℝ) (range ⇑b)) ∧ (convexHull ℝ) (range ⇑b) ⊆ s

Tactic:
obtain ⟨ε, hε, hεs⟩ := Metric.mem_nhds_iff.1 hs

Result:
case intro.intro.intro
E✝ : Type u_1
inst✝⁴ : NormedAddCommGroup E✝
inst✝³ : NormedSpace ℝ E✝
s✝ : Set E✝
x : E✝
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : FiniteDimensional ℝ E
s : Set E
hs : s ∈ 𝓝 0
b : AffineBasis (Fin (finrank ℝ E + 1)) ℝ E
c : AffineBasis (Fin (finrank ℝ E + 1)) ℝ E := -Finset.centroid ℝ Finset.univ ⇑b +ᵥ b
hc₀ : 0 ∈ interior ((convexHull ℝ) (range ⇑c))
cnorm : ℝ := Finset.univ.sup' ⋯ fun i => ‖c i‖
hcnorm : range ⇑c ⊆ closedBall 0 (cnorm + 1)
ε : ℝ
hε : ε > 0
hεs : ball 0 ε ⊆ s
⊢ ∃ b, 0 ∈ interior ((convexHull ℝ) (range ⇑b)) ∧ (convexHull ℝ) (range ⇑b) ⊆ s

Proof state:
case intro
E✝ : Type u_1
inst✝⁴ : NormedAddCommGroup E✝
inst✝³ : NormedSpace ℝ E✝
s✝ : Set E✝
x : E✝
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : FiniteDimensional ℝ E
s : Set E
hs : s ∈ 𝓝 0
b : AffineBasis (Fin (finrank ℝ E + 1)) ℝ E
c : AffineBasis (Fin (finrank ℝ E + 1)) ℝ E := -Finset.centroid ℝ Finset.univ ⇑b +ᵥ b
hc₀ : 0 ∈ interior ((convexHull ℝ) (range ⇑c))
cnorm : ℝ := Finset.univ.sup' ⋯ fun i => ‖c i‖
⊢ ∃ b, 0 ∈ interior ((convexHull ℝ) (range ⇑b)) ∧ (convexHull ℝ) (range ⇑b) ⊆ s

Tactic:
have hcnorm : range c ⊆ closedBall 0 (cnorm + 1) := by
  simpa only [cnorm, subset_def, Finset.mem_coe, mem_closedBall, dist_zero_right,
    ← sub_le_iff_le_add, Finset.le_sup'_iff, forall_mem_range] using fun i ↦ ⟨i, by simp⟩

Result:
case intro
E✝ : Type u_1
inst✝⁴ : NormedAddCommGroup E✝
inst✝³ : NormedSpace ℝ E✝
s✝ : Set E✝
x : E✝
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : FiniteDimensional ℝ E
s : Set E
hs : s ∈ 𝓝 0
b : AffineBasis (Fin (finrank ℝ E + 1)) ℝ E
c : AffineBasis (Fin (finrank ℝ E + 1)) ℝ E := -Finset.centroid ℝ Finset.univ ⇑b +ᵥ b
hc₀ : 0 ∈ interior ((convexHull ℝ) (range ⇑c))
cnorm : ℝ := Finset.univ.sup' ⋯ fun i => ‖c i‖
hcnorm : range ⇑c ⊆ closedBall 0 (cnorm + 1)
⊢ ∃ b, 0 ∈ interior ((convexHull ℝ) (range ⇑b)) ∧ (convexHull ℝ) (range ⇑b) ⊆ s

Proof state:
case intro
E✝ : Type u_1
inst✝⁴ : NormedAddCommGroup E✝
inst✝³ : NormedSpace ℝ E✝
s✝ : Set E✝
x : E✝
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : FiniteDimensional ℝ E
s : Set E
hs : s ∈ 𝓝 0
b : AffineBasis (Fin (finrank ℝ E + 1)) ℝ E
c : AffineBasis (Fin (finrank ℝ E + 1)) ℝ E := -Finset.centroid ℝ Finset.univ ⇑b +ᵥ b
hc₀ : 0 ∈ interior ((convexHull ℝ) (range ⇑c))
⊢ ∃ b, 0 ∈ interior ((convexHull ℝ) (range ⇑b)) ∧ (convexHull ℝ) (range ⇑b) ⊆ s

Tactic:
set cnorm := Finset.univ.sup' Finset.univ_nonempty (fun i ↦ ‖c i‖)

Result:
case intro
E✝ : Type u_1
inst✝⁴ : NormedAddCommGroup E✝
inst✝³ : NormedSpace ℝ E✝
s✝ : Set E✝
x : E✝
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : FiniteDimensional ℝ E
s : Set E
hs : s ∈ 𝓝 0
b : AffineBasis (Fin (finrank ℝ E + 1)) ℝ E
c : AffineBasis (Fin (finrank ℝ E + 1)) ℝ E := -Finset.centroid ℝ Finset.univ ⇑b +ᵥ b
hc₀ : 0 ∈ interior ((convexHull ℝ) (range ⇑c))
cnorm : ℝ := Finset.univ.sup' ⋯ fun i => ‖c i‖
⊢ ∃ b, 0 ∈ interior ((convexHull ℝ) (range ⇑b)) ∧ (convexHull ℝ) (range ⇑b) ⊆ s