| import Mathlib |
| import OAI.Geometry.SmoothYau.Smoothness.GlobalPrepDualNorm |
| import OAI.Geometry.SmoothYau.Smoothness.SmDualNorm |
|
|
| namespace OAI |
|
|
| noncomputable section |
| namespace YauCounterexamples |
| section |
| open Set Filter Function Metric |
| open scoped Topology ContDiff InnerProductSpace |
| variable {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E] [FiniteDimensional ℝ E] |
| local instance perturbDualNorm : NormedAddCommGroup (E →L[ℝ] ℝ) := inferInstance |
| local instance perturbDualSpace : NormedSpace ℝ (E →L[ℝ] ℝ) := inferInstance |
| local instance perturbFormNorm : NormedAddCommGroup (CoordinateForm E) := inferInstance |
| local instance perturbFormSpace : NormedSpace ℝ (CoordinateForm E) := inferInstance |
| lemma actualCoordinateHessian_const_mul (g : SmoothMetric E E) {f : E → ℝ} |
| (hf : ContDiff ℝ ∞ f) (s : ℝ) (x v w : E) : |
| actualCoordinateHessian g (fun y => s*f y) x v w = s*actualCoordinateHessian g f x v w := by |
| let L : ℝ →L[ℝ] ℝ := s • ContinuousLinearMap.id ℝ ℝ |
| change actualCoordinateHessian g (fun y => L (f y)) x v w = _ |
| rw [actualCoordinateHessian_apply,fderiv_second_clm_comp_apply L hf, |
| fderiv_clm_comp_apply L hf,actualCoordinateHessian_apply] |
| simp only [L,smul_apply,ContinuousLinearMap.id_apply,smul_eq_mul] |
| ring |
|
|
| theorem exists_compact_strict_perturbation (g : SmoothMetric E E) {f h : E → ℝ} |
| (hf : ContDiff ℝ ∞ f) (hh : ContDiff ℝ ∞ h) {K : Set E} (hK : IsCompact K) |
| (hstrict : ∀ x ∈ K, coordinateMetricGradient g f x ≠ 0 ∧ actualProfileStrict g f x) : |
| ∃ ε>0, ∀ s : ℝ, |s|<ε → ∀ x ∈ K, |
| coordinateMetricGradient g (fun y => f y+s*h y) x ≠ 0 ∧ |
| actualProfileStrict g (fun y => f y+s*h y) x := by |
| let J : E × ℝ → MetricJet E := fun z => |
| (z.1,actualCoordinateHessian g f z.1+z.2 • actualCoordinateHessian g h z.1, |
| fderiv ℝ f z.1+z.2 • fderiv ℝ h z.1) |
| have hHf : Continuous (actualCoordinateHessian g f) := |
| continuous_coordinateCovariantSecond _ (contDiff_metricChristoffel g).continuous hf |
| have hHh : Continuous (actualCoordinateHessian g h) := |
| continuous_coordinateCovariantSecond _ (contDiff_metricChristoffel g).continuous hh |
| have hJ : Continuous J := continuous_fst.prodMk |
| (((hHf.comp continuous_fst).add (continuous_snd.smul (hHh.comp continuous_fst))).prodMk |
| (((hf.continuous_fderiv (by simp)).comp continuous_fst).add |
| (continuous_snd.smul ((hh.continuous_fderiv (by simp)).comp continuous_fst)))) |
| have hsub : K ×ˢ ({0}:Set ℝ) ⊆ J ⁻¹' {z | metricJetStrict g z} := by |
| rintro ⟨x,s⟩ ⟨hx,hs⟩ |
| simp only [mem_singleton_iff] at hs |
| subst s |
| simpa only [mem_preimage,mem_ofPred_eq,J,zero_smul,add_zero,metricJetStrict_actual] using hstrict x hx |
| obtain ⟨U,V,hU,hV,hKU,h0V,hUV⟩ := generalized_tube_lemma hK isCompact_singleton |
| ((isOpen_metricJetStrict g).preimage hJ) hsub |
| obtain ⟨ε,hε,hεV⟩ := Metric.mem_nhds_iff.mp (hV.mem_nhds (h0V (mem_singleton 0))) |
| refine ⟨ε,hε,?_⟩ |
| intro s hs x hx |
| have hj := hUV (show (x,s) ∈ U ×ˢ V from |
| ⟨hKU hx,hεV (by simpa only [mem_ball,dist_zero_right,Real.norm_eq_abs] using hs)⟩) |
| have heH : actualCoordinateHessian g (fun y => f y+s*h y) x = |
| actualCoordinateHessian g f x+s • actualCoordinateHessian g h x := by |
| ext v w |
| rw [actualCoordinateHessian_add g hf (contDiff_const.mul hh), |
| actualCoordinateHessian_const_mul g hh] |
| rfl |
| have heD : fderiv ℝ (fun y => f y+s*h y) x = fderiv ℝ f x+s • fderiv ℝ h x := by |
| rw [fderiv_fun_add (hf.differentiable (by simp) x) |
| ((contDiff_const.mul hh).differentiable (by simp) x), |
| fderiv_const_mul (hh.differentiable (by simp) x)] |
| rw [← metricJetStrict_actual,heH,heD] |
| exact hj |
| end |
|
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|
|
| open Set Filter Function Metric |
| open scoped Topology ContDiff InnerProductSpace |
| variable {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E] [FiniteDimensional ℝ E] |
|
|
| theorem exists_compact_signed_envelope (g : SmoothMetric E E) {f l : E → ℝ} |
| (hf : ContDiff ℝ ∞ f) {R : ℝ} (hR : 0<R) (C : ℝ) |
| (hmin : ∀ x, C+1≤l x) (heq : ∀ x ∈ closedBall 0 R, f x=l x) |
| (hstrict : ∀ x ∈ closedBall 0 R, |
| coordinateMetricGradient g f x ≠ 0 ∧ actualProfileStrict g f x) : |
| ∃ δ>0, δ≤1/2 ∧ ∃ φ : E → ℝ, ContDiff ℝ ∞ φ ∧ |
| HasCompactSupport (fun x => φ x-C) ∧ |
| tsupport (fun x => φ x-C) ⊆ closedBall 0 (3*R/4) ∧ |
| (∀ x ∈ closedBall 0 (R/3), |
| coordinateMetricGradient g φ x ≠ 0 ∧ actualProfileStrict g φ x) ∧ |
| (∀ x ∈ closedBall 0 (R/8), φ x=l x+δ) ∧ |
| (∀ x ∉ closedBall 0 (R/3), φ x≤l x-δ) := by |
| let β : ContDiffBump (0:E) := ⟨R/8,R/4,by positivity,by linarith⟩ |
| let χ : ContDiffBump (0:E) := ⟨R/2,3*R/4,by positivity,by linarith⟩ |
| let h : E → ℝ := fun x => 2*β x-1 |
| have hh : ContDiff ℝ ∞ h := (contDiff_const.mul β.contDiff).sub contDiff_const |
| obtain ⟨ε,hε,hstable⟩ := exists_compact_strict_perturbation g hf hh |
| (isCompact_closedBall 0 (R/3)) (fun x hx => hstrict x |
| (closedBall_subset_closedBall (by linarith) hx)) |
| let δ := min (ε/2) (1/2) |
| have hδ : 0<δ := lt_min (half_pos hε) (by norm_num) |
| have hδε : |δ|<ε := by rw [abs_of_pos hδ]; exact (min_le_left _ _).trans_lt (half_lt_self hε) |
| have hδ1 : δ≤1/2 := min_le_right _ _ |
| let φ : E → ℝ := fun x => C+χ x*(f x-δ-C)+2*δ*β x |
| have hφ : ContDiff ℝ ∞ φ := |
| (contDiff_const.add (χ.contDiff.mul ((hf.sub contDiff_const).sub contDiff_const))).add |
| (contDiff_const.mul β.contDiff) |
| have hcover : tsupport (fun x => φ x-C) ⊆ closedBall 0 (3*R/4) := by |
| apply closure_minimal ?_ isClosed_closedBall |
| intro x hx |
| by_contra hh |
| have hχ : χ x=0 := χ.zero_of_le_dist (by |
| have hn : 3*R/4 < dist x 0 := by simpa only [mem_closedBall,not_le] using hh |
| exact hn.le) |
| have hβ : β x=0 := β.zero_of_le_dist (by |
| have hn : 3*R/4 < dist x 0 := by simpa only [mem_closedBall,not_le] using hh |
| dsimp [β]; linarith) |
| exact hx (by simp [φ,hχ,hβ]) |
| refine ⟨δ,hδ,hδ1,φ,hφ,?_,hcover,?_,?_,?_⟩ |
| · exact (isCompact_closedBall 0 (3*R/4)).of_isClosed_subset (isClosed_tsupport _) hcover |
| · intro x hx |
| have heχ := χ.eventuallyEq_one_of_mem_ball (show x ∈ ball 0 χ.rIn from |
| closedBall_subset_ball (by dsimp [χ]; linarith) hx) |
| have he : φ =ᶠ[𝓝 x] (fun y => f y+δ*h y) := by |
| filter_upwards [heχ] with y hy |
| dsimp [φ,h] |
| change χ y=1 at hy |
| rw [hy] |
| ring |
| have hj := actualJet_eventuallyEq g he |
| rw [hj.1,hj.2.2.1] |
| exact hstable δ hδε x hx |
| · intro x hx |
| have hβ := β.one_of_mem_closedBall hx |
| have hχ := χ.one_of_mem_closedBall (closedBall_subset_closedBall (by dsimp [χ]; linarith) hx) |
| have he := heq x (closedBall_subset_closedBall (by linarith) hx) |
| dsimp [φ] |
| rw [hβ,hχ,he] |
| ring |
| · intro x hx |
| have hd : R/3 < dist x 0 := by simpa only [mem_closedBall,not_le] using hx |
| have hβ : β x=0 := β.zero_of_le_dist (by dsimp [β]; linarith) |
| have hlow : C≤l x-δ := by have hm:=hmin x; linarith |
| dsimp [φ] |
| rw [hβ,mul_zero,add_zero] |
| by_cases hz : χ x=0 |
| · simpa [hz] using hlow |
| · have hmem : x ∈ ball 0 χ.rOut := by rwa [←χ.support_eq] |
| have he := heq x (ball_subset_closedBall.trans |
| (closedBall_subset_closedBall (by dsimp [χ]; linarith)) hmem) |
| rw [he] |
| have hc := χ.le_one (x:=x) |
| have hn := χ.nonneg (x:=x) |
| nlinarith |
|
|
| end YauCounterexamples |
| end |
|
|
| end OAI |
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