| import Mathlib |
| import OAI.Geometry.SmoothYau.Smoothness.BoundedDegreeMonomials |
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|
| namespace OAI |
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| noncomputable section |
| open Set Filter |
| open scoped Topology ContDiff |
| open Set Filter |
| open scoped Topology ContDiff |
| open MvPolynomial |
| open Set Filter |
| open scoped ContDiff |
| open Set Filter |
| open scoped Topology ContDiff |
| open Set Filter MvPolynomial |
| open scoped Topology ContDiff |
| open Set Filter Function MvPolynomial |
| open scoped Topology ContDiff |
| open Set Filter Function MvPolynomial |
| open scoped Topology ContDiff |
| open Set Filter |
| open scoped Topology ContDiff |
| namespace YauCounterexamples |
| variable {E X : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] |
| [FiniteDimensional ℝ E] [TopologicalSpace X] |
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| omit [FiniteDimensional ℝ E] in |
| lemma finiteTaylorFunction_three (f : E → ℝ) (x : E) : |
| finiteTaylorFunction f 3 x = f 0 + fderiv ℝ f 0 x + |
| (2 : ℝ)⁻¹ * iteratedFDeriv ℝ 2 f 0 (fun _ => x) := by |
| simp [finiteTaylorFunction, Finset.sum_range_succ, iteratedFDeriv_one_apply, |
| Nat.factorial, smul_eq_mul] |
|
|
| theorem compact_uniform_negative_hessian_localization |
| (f : X → E → ℝ) (hf : ∀ p, ContDiff ℝ ∞ (f p)) |
| (hthird : Continuous (fun q : X × E => iteratedFDeriv ℝ 3 |
| (fun y => f q.1 y - finiteTaylorFunction (f q.1) 3 y) q.2)) |
| {K : Set X} (hK : IsCompact K) (δ : X → ℝ) (κ : ℝ) (hκ : 0 < κ) |
| (hlinear : ∀ p ∈ K, ∀ x, fderiv ℝ (f p) 0 x ≤ δ p * ‖x‖) |
| (hquadratic : ∀ p ∈ K, ∀ x, |
| iteratedFDeriv ℝ 2 (f p) 0 (fun _ => x) ≤ -4 * κ * ‖x‖ ^ 2) : |
| ∃ r : ℝ, 0 < r ∧ ∀ p ∈ K, ∀ x ∈ Metric.ball (0 : E) r, |
| f p x ≤ f p 0 + δ p * ‖x‖ - κ * ‖x‖ ^ 2 := by |
| obtain ⟨C₀,hC₀⟩ := (hK.prod (isCompact_closedBall (0 : E) 1)).exists_bound_of_continuousOn |
| hthird.continuousOn |
| let C := max C₀ 1 |
| have hC : 0 < C := lt_of_lt_of_le zero_lt_one (le_max_right _ _) |
| refine ⟨min 1 (κ / C), lt_min zero_lt_one (div_pos hκ hC), ?_⟩ |
| intro p hp x hx |
| have hbound := vanishing_jet_norm_bound |
| (fun y => f p y - finiteTaylorFunction (f p) 3 y) |
| ((hf p).sub (contDiff_finiteTaylorFunction (f p) 3)) 3 1 C hC.le |
| (fun k hk => finiteTaylorFunction_remainder_jet (hf p) 3 k hk) |
| (fun y hy => (hC₀ (p,y) ⟨hp,Metric.ball_subset_closedBall hy⟩).trans (le_max_left _ _)) |
| have hxnorm : ‖x‖ < min 1 (κ / C) := by simpa using hx |
| have hxball : x ∈ Metric.ball (0 : E) 1 := by |
| simpa using hxnorm.trans_le (min_le_left _ _) |
| have hxsmall : C * ‖x‖ ≤ κ := by |
| have h := (lt_div_iff₀ hC).mp (hxnorm.trans_le (min_le_right _ _)) |
| nlinarith |
| have herr := hbound 3 (le_refl _) x hxball |
| change ‖iteratedFDeriv ℝ 0 (fun y => f p y - finiteTaylorFunction (f p) 3 y) x‖ ≤ |
| C * ‖x‖ ^ 3 at herr |
| rw [norm_iteratedFDeriv_zero, Real.norm_eq_abs] at herr |
| have hu : f p x - finiteTaylorFunction (f p) 3 x ≤ C * ‖x‖ ^ 3 := |
| (le_abs_self _).trans herr |
| rw [finiteTaylorFunction_three] at hu |
| have hq := hquadratic p hp x |
| have hl := hlinear p hp x |
| have hrem : C * ‖x‖ ^ 3 ≤ κ * ‖x‖ ^ 2 := by |
| have h := mul_le_mul_of_nonneg_right hxsmall (sq_nonneg ‖x‖) |
| nlinarith |
| norm_num at hu |
| linarith |
| end YauCounterexamples |
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| end |
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| end OAI |
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