/-- If a function is differentiable inside a convex set with nonempty interior, and has a second
derivative at a point of this convex set, then this second derivative is symmetric. -/
theorem Convex.second_derivative_within_at_symmetric {s : Set E} (s_conv : Convex ℝ s)
    (hne : (interior s).Nonempty) {f : E → F} {f' : E → E →L[ℝ] F} {f'' : E →L[ℝ] E →L[ℝ] F}
    (hf : ∀ x ∈ interior s, HasFDerivAt f (f' x) x) {x : E} (xs : x ∈ s)
    (hx : HasFDerivWithinAt f' f'' (interior s) x) (v w : E) : f'' v w = f'' w v := by
  /- we work around a point `x + 4 z` in the interior of `s`. For any vector `m`,
    then `x + 4 (z + t m)` also belongs to the interior of `s` for small enough `t`. This means that
    we will be able to apply `second_derivative_within_at_symmetric_of_mem_interior` to show
    that `f''` is symmetric, after cancelling all the contributions due to `z`. -/
  rcases hne with ⟨y, hy⟩
  obtain ⟨z, hz⟩ : ∃ z, z = ((1 : ℝ) / 4) • (y - x) := ⟨((1 : ℝ) / 4) • (y - x), rfl⟩
  have A : ∀ m : E, Filter.Tendsto (fun t : ℝ => x + (4 : ℝ) • (z + t • m)) (𝓝 0) (𝓝 y) := by
    intro m
    have : x + (4 : ℝ) • (z + (0 : ℝ) • m) = y := by simp [hz]
    rw [← this]
    refine' tendsto_const_nhds.add <| tendsto_const_nhds.smul <| tendsto_const_nhds.add _
    exact continuousAt_id.smul continuousAt_const
  have B : ∀ m : E, ∀ᶠ t in 𝓝[>] (0 : ℝ), x + (4 : ℝ) • (z + t • m) ∈ interior s := by
    intro m
    apply nhdsWithin_le_nhds
    apply A m
    rw [mem_interior_iff_mem_nhds] at hy
    exact interior_mem_nhds.2 hy
  -- we choose `t m > 0` such that `x + 4 (z + (t m) m)` belongs to the interior of `s`, for any
  -- vector `m`.
  choose t ts tpos using fun m => ((B m).and self_mem_nhdsWithin).exists
  -- applying `second_derivative_within_at_symmetric_of_mem_interior` to the vectors `z`
  -- and `z + (t m) m`, we deduce that `f'' m z = f'' z m` for all `m`.
  have C : ∀ m : E, f'' m z = f'' z m := by
    intro m
    have : f'' (z + t m • m) (z + t 0 • (0 : E)) = f'' (z + t 0 • (0 : E)) (z + t m • m) :=
      s_conv.second_derivative_within_at_symmetric_of_mem_interior hf xs hx (ts 0) (ts m)
    simp only [ContinuousLinearMap.map_add, ContinuousLinearMap.map_smul, add_right_inj,
      ContinuousLinearMap.add_apply, Pi.smul_apply, ContinuousLinearMap.coe_smul', add_zero,
      ContinuousLinearMap.zero_apply, smul_zero, ContinuousLinearMap.map_zero] at this
    exact smul_right_injective F (tpos m).ne' this
  -- applying `second_derivative_within_at_symmetric_of_mem_interior` to the vectors `z + (t v) v`
  -- and `z + (t w) w`, we deduce that `f'' v w = f'' w v`. Cross terms involving `z` can be
  -- eliminated thanks to the fact proved above that `f'' m z = f'' z m`.
  

Proof state:
case intro.intro
E : Type u_1
F : Type u_2
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
s✝ : Set E
s_conv✝ : Convex ℝ s✝
f✝ : E → F
f'✝ : E → E →L[ℝ] F
f''✝ : E →L[ℝ] E →L[ℝ] F
hf✝ : ∀ x ∈ interior s✝, HasFDerivAt f✝ (f'✝ x) x
x✝ : E
xs✝ : x✝ ∈ s✝
hx✝ : HasFDerivWithinAt f'✝ f''✝ (interior s✝) x✝
s : Set E
s_conv : Convex ℝ s
f : E → F
f' : E → E →L[ℝ] F
f'' : E →L[ℝ] E →L[ℝ] F
hf : ∀ x ∈ interior s, HasFDerivAt f (f' x) x
x : E
xs : x ∈ s
hx : HasFDerivWithinAt f' f'' (interior s) x
v w y : E
hy : y ∈ interior s
z : E
hz : z = (1 / 4) • (y - x)
A : ∀ (m : E), Filter.Tendsto (fun t => x + 4 • (z + t • m)) (𝓝 0) (𝓝 y)
B : ∀ (m : E), ∀ᶠ (t : ℝ) in 𝓝[>] 0, x + 4 • (z + t • m) ∈ interior s
t : E → ℝ
ts : ∀ (m : E), x + 4 • (z + t m • m) ∈ interior s
tpos : ∀ (m : E), 0 < t m
C : ∀ (m : E), (f'' m) z = (f'' z) m
⊢ (f'' v) w = (f'' w) v

Next tactic:
have : f'' (z + t v • v) (z + t w • w) = f'' (z + t w • w) (z + t v • v) :=
    s_conv.second_derivative_within_at_symmetric_of_mem_interior hf xs hx (ts w) (ts v)