| import Mathlib |
| import OAI.Geometry.CubicTorus.Profiles.NestingEndpointBound |
|
|
| namespace OAI |
|
|
| section |
| section |
| noncomputable section |
| open MeasureTheory Set Filter |
| open scoped ENNReal NNReal Topology RealInnerProductSpace |
| namespace CubicTorus |
|
|
| theorem exists_smooth_seam_probability {L : ℝ} (hL : L∈Ioo 0 (1/2)) : |
| ∃ β : ℝ →ℝ,ContDiff ℝ (⊤ : ℕ∞) β ∧Function.Periodic β 1 ∧ |
| (∀ t,0≤β t) ∧(∫ t in (0 : ℝ)..1,β t)=1 ∧ |
| (∀ t,|t|≤L →β t=0) ∧ |
| (∀ t : ℝ,β t≠0 →dist (t : AxisCircle) ((1/2 : ℝ) : AxisCircle)<1/2-L) := by |
| let r : ℝ := 1/2-L |
| have hr : 0<r := by dsimp only [r]; linarith [hL.2] |
| let b : ContDiffBump (0 : ℝ) := ⟨r/2,r,half_pos hr,by linarith⟩ |
| let B : ℝ →ℝ := fun t ↦ b.normed volume (t-1/2) |
| have hB : ContDiff ℝ (⊤ : ℕ∞) B := b.contDiff_normed.comp (contDiff_id.sub contDiff_const) |
| have hBs : tsupport B ⊆ Ioo ((1/2 : ℝ)-1/2) (1/2+1/2) := by |
| intro t ht |
| have hh := tsupport_comp_subset_preimage (b.normed volume) (continuous_id.sub continuous_const) ht |
| change t-1/2∈tsupport (b.normed volume) at hh |
| rw [b.tsupport_normed_eq,Real.closedBall_eq_Icc] at hh |
| change 0-r≤t-1/2 ∧ t-1/2≤0+r at hh |
| dsimp only [r] at hh |
| constructor <;> linarith [hh.1,hh.2,hL.1] |
| have hBc : HasCompactSupport B := IsCompact.of_isClosed_subset isCompact_Icc (isClosed_tsupport B) |
| (fun _ ht ↦ ⟨(hBs ht).1.le,(hBs ht).2.le⟩) |
| let β := circlePeriodize B |
| have hβ : ContDiff ℝ (⊤ : ℕ∞) β := circlePeriodize_contDiff hB hBc |
| have pβ : Function.Periodic β 1 := circlePeriodize_periodic B |
| have he (t : ℝ) (ht : t∈Icc (0 : ℝ) 1) : β t=B t := |
| (circlePeriodize_eventuallyEq hBc hBs (by convert ht using 1; norm_num)).eq_of_nhds |
| have hnB (t : ℝ) (ht : B t≠0) : |t-1/2|<r := by |
| have hh : t-1/2∈Metric.ball (0 : ℝ) r := by |
| rw [←b.support_normed_eq (μ := volume)] |
| exact ht |
| simpa only [Metric.mem_ball,Real.dist_eq,sub_zero] using hh |
| have hmean : (∫ t : AxisCircle,pβ.lift t)=1 := by |
| rw [circlePeriodize_integral_short hBc hBs] |
| exact (integral_sub_right_eq_self (b.normed volume) (1/2)).trans b.integral_normed |
| refine ⟨β,hβ,pβ,fun t ↦ tsum_nonneg (fun n : ℤ ↦ b.nonneg_normed (t+n-1/2)),?_,?_,?_⟩ |
| · have hh := AddCircle.intervalIntegral_preimage (1 : ℝ) (0 : ℝ) pβ.lift |
| simpa only [zero_add,pβ.lift_coe,hmean] using hh |
| · intro t ht |
| rcases le_total (0 : ℝ) t with ht0 | ht0 |
| · rw [he t ⟨ht0,by linarith [(abs_le.mp ht).2,hL.2]⟩] |
| by_contra hh |
| have hb := hnB t hh |
| dsimp only [r] at hb |
| have ha := (abs_lt.mp hb).1 |
| linarith [(abs_le.mp ht).2] |
| · rw [←pβ t,he (t+1) ⟨by linarith [(abs_le.mp ht).1,hL.2],by linarith⟩] |
| by_contra hh |
| have hb := hnB (t+1) hh |
| dsimp only [r] at hb |
| have ha := (abs_lt.mp hb).2 |
| linarith [(abs_le.mp ht).1] |
| · intro t ht |
| let x : ℝ := ((AddCircle.equivIco (1 : ℝ) 0) (t : AxisCircle) : ℝ) |
| have hx := ((AddCircle.equivIco (1 : ℝ) 0) (t : AxisCircle)).property |
| change 0≤x ∧ x<0+1 at hx |
| have hxt : (x : AxisCircle)=(t : AxisCircle) := (AddCircle.equivIco (1 : ℝ) 0).symm_apply_apply _ |
| have hβx : β x≠0 := by |
| have heq : β x=β t := by rw [←pβ.lift_coe x,←pβ.lift_coe t,hxt] |
| rwa [heq] |
| rw [he x ⟨hx.1,by linarith [hx.2]⟩] at hβx |
| have hnorm : dist (x : AxisCircle) ((1/2 : ℝ) : AxisCircle)≤|x-1/2| := by |
| rw [dist_eq_norm,←QuotientAddGroup.mk_sub] |
| exact QuotientAddGroup.norm_mk_le_norm |
| rw [←hxt] |
| exact hnorm.trans_lt (hnB x hβx) |
|
|
| end CubicTorus |
|
|
| end |
| end |
|
|
| section |
|
|
| noncomputable section |
| open MeasureTheory Set Filter |
| open scoped ENNReal NNReal Topology RealInnerProductSpace |
| namespace CubicTorus |
|
|
| theorem exists_smooth_saw_primitive {β : ℝ →ℝ} |
| (hβ : ContDiff ℝ (⊤ : ℕ∞) β) (pβ : Function.Periodic β 1) |
| (hβ0 : ∀ t,0≤β t) (hβ1 : (∫ t in (0 : ℝ)..1,β t)=1) |
| {L k : ℝ} (hL : L∈Ioo 0 (1/2)) (hk : 0≤k) |
| (hb : ∀ t,|t|≤L →β t=0) : |
| ∃ v : ℝ →ℝ,ContDiff ℝ (⊤ : ℕ∞) v ∧Function.Periodic v 1 ∧ |
| (∀ t,deriv v t=2*k*(1-β t)) ∧ |
| (∀ t,|t|≤L →v t=2*k*t) ∧(∀ t,|v t|≤2*k*(1-L)) := by |
| obtain ⟨ψ,hψ,pψ,hdψ⟩ := smooth_periodic_centered_primitive hβ pβ |
| let v : ℝ →ℝ := fun t ↦ -2*k*(ψ t-ψ 0) |
| have hv : ContDiff ℝ (⊤ : ℕ∞) v := contDiff_const.mul (hψ.sub contDiff_const) |
| have pv : Function.Periodic v 1 := fun t ↦ by dsimp only [v]; rw [pψ t] |
| have hd (t : ℝ) : deriv v t=2*k*(1-β t) := by |
| dsimp only [v] |
| rw [deriv_const_mul _ ((hψ.differentiable (by simp) t).sub_const _),deriv_sub_const,hdψ,hβ1] |
| ring |
| have hcentral (t : ℝ) (ht : |t|≤L) : v t=2*k*t := by |
| have he : (∫ x in (0 : ℝ)..t,deriv v x)=2*k*t := by |
| calc |
| _=∫ x in (0 : ℝ)..t,(2*k : ℝ) := by |
| apply intervalIntegral.integral_congr |
| intro x hx |
| have hxL : |x|≤L := by |
| rcases le_total (0 : ℝ) t with ht0 | ht0 |
| · rw [uIcc_of_le ht0] at hx |
| exact abs_le.mpr ⟨by linarith [hx.1,hL.1],hx.2.trans (le_trans (le_abs_self t) ht)⟩ |
| · rw [uIcc_of_ge ht0] at hx |
| exact abs_le.mpr ⟨by linarith [(abs_le.mp ht).1,hx.1],by linarith [hx.2,hL.1]⟩ |
| rw [hd,hb x hxL] |
| ring |
| _=2*k*t := by rw [intervalIntegral.integral_const]; simp only [sub_zero,smul_eq_mul]; ring |
| rw [intervalIntegral.integral_deriv_eq_sub (fun x _ ↦ hv.differentiable (by simp) x) |
| ((hv.continuous_deriv (by simp)).intervalIntegrable _ _)] at he |
| simpa only [v,sub_self,mul_zero,sub_zero] using he |
| have hm : Antitone (fun t ↦ v t-2*k*t) := by |
| apply antitone_of_deriv_nonpos (hv.differentiable (by simp) |>.sub (differentiable_id.const_mul (2*k))) |
| intro t |
| rw [deriv_sub (hv.differentiable (by simp) t) ((differentiable_id.const_mul (2*k)) t), |
| hd,deriv_const_mul _ (differentiableAt_id),deriv_id,mul_one] |
| nlinarith [hβ0 t,mul_nonneg hk (hβ0 t)] |
| have hs (a b : ℝ) (hab : a≤b) : v b-v a≤2*k*(b-a) := by |
| have hh := hm hab |
| dsimp only at hh |
| linarith |
| refine ⟨v,hv,pv,hd,hcentral,fun t ↦ ?_⟩ |
| let x : ℝ := ((AddCircle.equivIco (1 : ℝ) (-L)) (t : AxisCircle) : ℝ) |
| have hx := ((AddCircle.equivIco (1 : ℝ) (-L)) (t : AxisCircle)).property |
| change -L≤x ∧ x< -L+1 at hx |
| have hxt : (x : AxisCircle)=(t : AxisCircle) := |
| (AddCircle.equivIco (1 : ℝ) (-L)).symm_apply_apply _ |
| have hvt : v t=v x := by rw [←pv.lift_coe t,←pv.lift_coe x,hxt] |
| rw [hvt] |
| by_cases hxc : x≤L |
| · rw [hcentral x (abs_le.mpr ⟨hx.1,hxc⟩),abs_mul,abs_of_nonneg (by positivity : 0≤2*k)] |
| exact mul_le_mul_of_nonneg_left ((abs_le.mpr ⟨hx.1,hxc⟩).trans (by linarith [hL.2])) (by positivity) |
| · have hvL : v L=2*k*L := hcentral L (by rw [abs_of_pos hL.1]) |
| have hvR : v (1-L)= -2*k*L := by |
| rw [show 1-L= -L+1 by ring,pv (-L),hcentral (-L) (by rw [abs_neg,abs_of_pos hL.1])] |
| ring |
| have h1 := hs L x (le_of_not_ge hxc) |
| have h2 := hs x (1-L) (by linarith [hx.2,hL.1]) |
| rw [hvL] at h1 |
| rw [hvR] at h2 |
| apply abs_le.mpr |
| constructor |
| · have h := mul_le_mul_of_nonneg_left (show L≤x from (le_of_not_ge hxc)) (show 0≤2*k by positivity) |
| nlinarith |
| · have h := mul_le_mul_of_nonneg_left (show x≤1-L by linarith [hx.2,hL.1]) (show 0≤2*k by positivity) |
| nlinarith |
|
|
| end CubicTorus |
|
|
| end |
| end |
|
|
| section |
|
|
| noncomputable section |
| open MeasureTheory Set Filter |
| open scoped ENNReal NNReal Topology RealInnerProductSpace |
| namespace CubicTorus |
|
|
| theorem planarTorusPerimeter_ge_strip_calibration |
| {A : Set TransverseProduct} (hA : IsOpen A) |
| (hin : ∀ y : TransverseProduct,y 0=0 →y∈A) |
| (hout : ∀ y : TransverseProduct,y 0=((1/2 : ℝ) : AxisCircle) →y∉closure A) |
| {e k : ℝ} (he : 0≤e) (hk : k∈Ico 0 1) |
| (hedge : ∀ y∈closure A,y 1=((1/2 : ℝ) : AxisCircle) →2*‖y 0‖≤e) : |
| ENNReal.ofReal (2*k*(volume.real A-e)+ |
| 2*(∫ t in (-(1/2 : ℝ))..(1/2),Real.sqrt (1-(2*k*t)^2)))≤planarTorusPerimeter A := by |
| let f : ℝ →ℝ := fun t ↦ Real.sqrt (1-(2*k*t)^2) |
| have hf : Continuous f := (continuous_const.sub ((continuous_const.mul continuous_id).pow 2)).sqrt |
| have hp : Continuous (fun z : ℝ ↦ ∫ t in (0 : ℝ)..z,f t) := |
| intervalIntegral.continuous_primitive (fun _ _ ↦ hf.intervalIntegrable _ _) 0 |
| have hI : Continuous (fun L : ℝ ↦ ∫ t in (-L)..L,f t) := by |
| have heq (L : ℝ) : (∫ t in (-L)..L,f t)=(∫ t in (0 : ℝ)..L,f t)-(∫ t in (0 : ℝ)..(-L),f t) := |
| (intervalIntegral.integral_interval_sub_left (hf.intervalIntegrable _ _) (hf.intervalIntegrable _ _)).symm |
| simp_rw [heq] |
| exact hp.sub (hp.comp continuous_neg) |
| have happrox (ε : ℝ) (hε : 0<ε) : |
| ENNReal.ofReal (2*k*(volume.real A-(e+ε))+2*(∫ t in (-(1/2 : ℝ))..(1/2),f t))≤planarTorusPerimeter A := by |
| obtain ⟨δ,hδ,hcollar⟩ := planar_closed_edge_collar hedge hε |
| let U : Set ℝ := {L | 0<L ∧1/2-L<δ ∧2*k*(1-L)<1} |
| have hU : IsOpen U := (isOpen_lt continuous_const continuous_id).inter |
| ((isOpen_lt (continuous_const.sub continuous_id) continuous_const).inter |
| (isOpen_lt (continuous_const.mul (continuous_const.sub continuous_id)) continuous_const)) |
| have hhalf : (1/2 : ℝ)∈U := by |
| change 0<(1/2 : ℝ) ∧1/2-1/2<δ ∧2*k*(1-1/2)<1 |
| constructor |
| · norm_num |
| constructor |
| · simpa using hδ |
| · nlinarith [hk.2] |
| obtain ⟨η,hη,hηU⟩ := Metric.isOpen_iff.mp hU _ hhalf |
| have hfinite (L : ℝ) (hL : L∈Ioo (1/2-η) (1/2)) : |
| ENNReal.ofReal (2*k*(volume.real A-(e+ε))+2*(∫ t in (-L)..L,f t))≤planarTorusPerimeter A := by |
| have hLU : L∈U := hηU (show dist L (1/2 : ℝ)<η from by |
| rw [Real.dist_eq,abs_of_neg (by linarith [hL.2])]; linarith [hL.1]) |
| have hLL : L∈Ioo 0 (1/2) := ⟨hLU.1,hL.2⟩ |
| obtain ⟨β,hβ,pβ,hβ0,hβ1,hβcentral,hβsupport⟩ := exists_smooth_seam_probability hLL |
| obtain ⟨v,hv,pv,hd,hcentral,hbound⟩ := exists_smooth_saw_primitive hβ pβ hβ0 hβ1 hLL hk.1 hβcentral |
| have hvb (t : ℝ) : |v t|<1 := (hbound t).trans_lt hLU.2.2 |
| have hpos (t : ℝ) : 0<1-(v t)^2 := by |
| nlinarith [hvb t,sq_abs (v t),abs_nonneg (v t)] |
| let u : ℝ →ℝ := fun t ↦ Real.sqrt (1-(v t)^2) |
| have hu : ContDiff ℝ (⊤ : ℕ∞) u := (contDiff_const.sub (hv.pow 2)).sqrt (fun t ↦ (hpos t).ne') |
| have pu : Function.Periodic u 1 := fun t ↦ by dsimp only [u]; rw [pv t] |
| have huv (t : ℝ) : (u t)^2+(v t)^2≤1 := by |
| dsimp only [u] |
| rw [Real.sq_sqrt (hpos t).le] |
| linarith |
| have hβC : Continuous (pβ.lift : AxisCircle →ℝ) := continuous_periodic_lift pβ hβ.continuous |
| have hβC0 (t : AxisCircle) : 0≤pβ.lift t := by |
| induction t using Quotient.inductionOn with | h t => exact hβ0 t |
| have hβC1 : (∫ t : AxisCircle,pβ.lift t)=1 := by |
| rw [←AddCircle.intervalIntegral_preimage (1 : ℝ) (0 : ℝ)] |
| simpa only [zero_add,pβ.lift_coe] using hβ1 |
| have hβCS (t : AxisCircle) (ht : pβ.lift t≠0) : dist t ((1/2 : ℝ) : AxisCircle)<δ := by |
| induction t using Quotient.inductionOn with |
| | h t => exact (hβsupport t ht).trans hLU.2.1 |
| have hcal := planarTorusPerimeter_ge_strip_seam_calibration hA (closedCentralArc_isCompact L) |
| hu hv pu pv (fun t ↦ Real.sqrt_nonneg _) huv (fun y hy _ ↦ hin y hy) hout |
| hβC hβC0 hβC1 hk.1 (show 0≤e+ε by linarith) |
| (fun t ↦ hd t) (fun y hy hn ↦ (hcollar y (subset_closure hy) (hβCS (y 1) hn)).le) |
| have hi : (∫ t in closedCentralArc L,pu.lift t)=∫ t in (-L)..L,f t := by |
| rw [integral_closedCentralArc ⟨hLL.1.le,hLL.2⟩] |
| apply intervalIntegral.integral_congr |
| intro t ht |
| rw [uIcc_of_le (by linarith [hLL.1])] at ht |
| change Real.sqrt (1-(v t)^2)=Real.sqrt (1-(2*k*t)^2) |
| rw [hcentral t (abs_le.mpr ht)] |
| rwa [hi] at hcal |
| have hc : Continuous (fun L : ℝ ↦ ENNReal.ofReal (2*k*(volume.real A-(e+ε))+2*(∫ t in (-L)..L,f t))) := |
| ENNReal.continuous_ofReal.comp (continuous_const.add (continuous_const.mul hI)) |
| have hs : IsClosed {L : ℝ | ENNReal.ofReal (2*k*(volume.real A-(e+ε))+2*(∫ t in (-L)..L,f t))≤planarTorusPerimeter A} := |
| isClosed_le hc continuous_const |
| have hh : (1/2 : ℝ)∈closure (Ioo (1/2-η) (1/2)) := by |
| rw [closure_Ioo (by linarith : (1/2 : ℝ)-η≠1/2)] |
| exact ⟨by linarith,le_rfl⟩ |
| exact closure_minimal (fun L hL ↦ hfinite L hL) hs hh |
| have hc : Continuous (fun ε : ℝ ↦ ENNReal.ofReal |
| (2*k*(volume.real A-(e+ε))+2*(∫ t in (-(1/2 : ℝ))..(1/2),f t))) := |
| ENNReal.continuous_ofReal.comp |
| ((continuous_const.mul (continuous_const.sub (continuous_const.add continuous_id))).add continuous_const) |
| have hs : IsClosed {ε : ℝ | ENNReal.ofReal |
| (2*k*(volume.real A-(e+ε))+2*(∫ t in (-(1/2 : ℝ))..(1/2),f t))≤planarTorusPerimeter A} := |
| isClosed_le hc continuous_const |
| have h0 : (0 : ℝ)∈closure (Ioi (0 : ℝ)) := by rw [closure_Ioi]; exact (le_rfl : (0 : ℝ)≤0) |
| have hh := closure_minimal (fun ε hε ↦ happrox ε hε) hs h0 |
| change ENNReal.ofReal (2*k*(volume.real A-(e+0))+2*(∫ t in (-(1/2 : ℝ))..(1/2),f t))≤planarTorusPerimeter A at hh |
| simpa only [add_zero] using hh |
|
|
| end CubicTorus |
|
|
| end |
| end |
|
|
| section |
|
|
| noncomputable section |
| open MeasureTheory Set Filter |
| open scoped ENNReal NNReal Topology RealInnerProductSpace |
| namespace CubicTorus |
|
|
| lemma strip_sqrt_polynomial_lower {w : ℝ} (hw : w∈Icc 0 ((21/40 : ℝ)^2)) : |
| 1-w/2-w^2/5≤Real.sqrt (1-w) := by |
| have hw2 : w^2≤((21/40 : ℝ)^2)^2 := by nlinarith [mul_nonneg hw.1 (sub_nonneg.mpr hw.2)] |
| have hc : -3/20+w/5+w^2/25≤0 := by nlinarith [hw.2,hw2] |
| have hh := mul_nonpos_of_nonneg_of_nonpos (sq_nonneg w) hc |
| apply Real.le_sqrt_of_sq_le |
| nlinarith |
|
|
| lemma strip_round_integral_lower {k : ℝ} (hk : k∈Icc 0 (21/40 : ℝ)) : |
| 1-k^2/6-k^4/25≤∫ t in (-(1/2 : ℝ))..(1/2),Real.sqrt (1-(2*k*t)^2) := by |
| let f : ℝ →ℝ := fun t ↦ 1-(2*k*t)^2/2-(2*k*t)^4/5 |
| have hf : Continuous f := (continuous_const.sub (((continuous_const.mul continuous_id).pow 2).div_const 2)).sub |
| (((continuous_const.mul continuous_id).pow 4).div_const 5) |
| have hg : Continuous (fun t : ℝ ↦ Real.sqrt (1-(2*k*t)^2)) := |
| (continuous_const.sub ((continuous_const.mul continuous_id).pow 2)).sqrt |
| have hb := intervalIntegral.integral_mono_on (μ := volume) (show -(1/2 : ℝ)≤1/2 by norm_num) |
| (hf.intervalIntegrable _ _) (hg.intervalIntegrable _ _) (fun t ht ↦ ?_) |
| · have hpoly : f=(fun t : ℝ ↦ 1-(2*k^2)*t^2-((16/5)*k^4)*t^4) := by funext t; dsimp only [f]; ring |
| have h2 : IntervalIntegrable (fun t : ℝ ↦ (2*k^2)*t^2) volume (-(1/2 : ℝ)) (1/2) := |
| (show Continuous (fun t : ℝ ↦ (2*k^2)*t^2) by fun_prop).intervalIntegrable _ _ |
| have h4 : IntervalIntegrable (fun t : ℝ ↦ ((16/5)*k^4)*t^4) volume (-(1/2 : ℝ)) (1/2) := |
| (show Continuous (fun t : ℝ ↦ ((16/5)*k^4)*t^4) by fun_prop).intervalIntegrable _ _ |
| rw [hpoly,intervalIntegral.integral_sub (intervalIntegrable_const.sub h2) h4, |
| intervalIntegral.integral_sub intervalIntegrable_const h2, |
| intervalIntegral.integral_const,intervalIntegral.integral_const_mul,intervalIntegral.integral_const_mul, |
| integral_pow,integral_pow] at hb |
| norm_num at hb |
| nlinarith [hb] |
| · have hm : |2*k*t|≤21/40 := by |
| calc |
| _=2*k*|t| := by rw [abs_mul,abs_mul,abs_of_nonneg (by norm_num : (0 : ℝ)≤2),abs_of_nonneg hk.1] |
| _≤2*k*(1/2) := mul_le_mul_of_nonneg_left (abs_le.mpr ht) (mul_nonneg (by norm_num) hk.1) |
| _≤21/40 := by linarith [hk.2] |
| have hw : (2*k*t)^2∈Icc 0 ((21/40 : ℝ)^2) := ⟨sq_nonneg _,by nlinarith [sq_abs (2*k*t),abs_nonneg (2*k*t)]⟩ |
| have hh := strip_sqrt_polynomial_lower hw |
| convert hh using 1 |
| dsimp only [f] |
| ring |
|
|
| theorem planarTorusPerimeter_ge_strip_gain |
| {A : Set TransverseProduct} (hA : IsOpen A) |
| (hin : ∀ y : TransverseProduct,y 0=0 →y∈A) |
| (hout : ∀ y : TransverseProduct,y 0=((1/2 : ℝ) : AxisCircle) →y∉closure A) |
| {e j : ℝ} (he : 0≤e) (hj : j∈Icc 0 (21/100 : ℝ)) |
| (hedge : ∀ y∈closure A,y 1=((1/2 : ℝ) : AxisCircle) →2*‖y 0‖≤e) |
| (hgap : j≤volume.real A-e) : |
| ENNReal.ofReal (2*(1+(27/20)*j^2))≤planarTorusPerimeter A := by |
| let k : ℝ := (5/2)*j |
| have hk : k∈Icc 0 (21/40 : ℝ) := by dsimp only [k]; constructor <;> linarith [hj.1,hj.2] |
| have hcal := planarTorusPerimeter_ge_strip_calibration hA hin hout he ⟨hk.1,by linarith [hk.2]⟩ hedge |
| have hint := strip_round_integral_lower hk |
| have hj2 : j^2≤(21/100 : ℝ)^2 := by nlinarith [mul_nonneg hj.1 (sub_nonneg.mpr hj.2)] |
| have hq : 0≤13/120-(25/16)*j^2 := by nlinarith [hj2] |
| have hqq := mul_nonneg (sq_nonneg j) hq |
| have ha := mul_le_mul_of_nonneg_left hgap hk.1 |
| apply le_trans (ENNReal.ofReal_le_ofReal ?_) hcal |
| dsimp only [k] at hint ha ⊢ |
| nlinarith |
|
|
| end CubicTorus |
|
|
| end |
| end |
|
|
| section |
|
|
| noncomputable section |
| open MeasureTheory Set Filter |
| open scoped ENNReal NNReal Topology RealInnerProductSpace |
| namespace CubicTorus |
|
|
| lemma planar_radial_strip_gain_first {A : Set TransverseProduct} (hA : IsOpen A) |
| (hd : PlanarRadialDownset A) (hin : planarAxisPoint (1/2)∈A) |
| (hout : planarFirstAxisPoint (1/2)∉closure A) |
| {e j : ℝ} (he : 0≤e) (hj : j∈Icc 0 (21/100 : ℝ)) |
| (hedge : ∀ y∈closure A,y 1=((1/2 : ℝ) : AxisCircle) →2*‖y 0‖≤e) |
| (hgap : j≤volume.real A-e) : |
| ENNReal.ofReal (2*(1+(27/20)*j^2))≤planarTorusPerimeter A := by |
| apply planarTorusPerimeter_ge_strip_gain hA ?_ (planar_radial_empty_fiber hd hout) he hj hedge hgap |
| intro y hy |
| apply hd y (planarAxisPoint (1/2)) ?_ hin |
| intro k |
| fin_cases k |
| · simp [hy,planarAxisPoint] |
| · change ‖y 1‖≤‖((1/2 : ℝ) : AxisCircle)‖ |
| rw [axisCircle_norm_coe_small (by norm_num : |(1/2 : ℝ)|≤1/2),abs_of_pos (by norm_num : (0 : ℝ)<1/2)] |
| exact axisCircle_norm_le_half (y 1) |
|
|
| lemma planar_radial_strip_gain_second {A : Set TransverseProduct} (hA : IsOpen A) |
| (hd : PlanarRadialDownset A) (hin : planarFirstAxisPoint (1/2)∈A) |
| (hout : planarAxisPoint (1/2)∉closure A) |
| {e j : ℝ} (he : 0≤e) (hj : j∈Icc 0 (21/100 : ℝ)) |
| (hedge : ∀ y∈closure A,y 0=((1/2 : ℝ) : AxisCircle) →2*‖y 1‖≤e) |
| (hgap : j≤volume.real A-e) : |
| ENNReal.ofReal (2*(1+(27/20)*j^2))≤planarTorusPerimeter A := by |
| have hi : planarAxisPoint (1/2)∈planarSwap ⁻¹' A := by simpa using hin |
| have ho : planarFirstAxisPoint (1/2)∉closure (planarSwap ⁻¹' A) := by |
| intro hh |
| apply hout |
| simpa only [mem_preimage,planarSwap_firstAxisPoint] using planarSwap_closure_subset A hh |
| have hb : ∀ y∈closure (planarSwap ⁻¹' A),y 1=((1/2 : ℝ) : AxisCircle) →2*‖y 0‖≤e := by |
| intro y hy hy1 |
| have hh := hedge (planarSwap y) (planarSwap_closure_subset A hy) (by simpa using hy1) |
| simpa using hh |
| have hh := planar_radial_strip_gain_first (hA.preimage planarSwap_continuous) |
| (planarRadialDownset_swap hd) hi ho he hj hb (by simpa only [planar_volume_swap] using hgap) |
| simpa only [planarTorusPerimeter_swap] using hh |
|
|
| end CubicTorus |
|
|
| end |
| end |
|
|
| section |
|
|
| noncomputable section |
| open MeasureTheory Set Filter |
| open scoped ENNReal NNReal Topology RealInnerProductSpace |
| namespace CubicTorus |
|
|
| theorem isMinimizerAt_section_strip_gain {V : ℝ} (hV : V∈Ioo 0 1) |
| {E : Set Torus} (hmin : IsMinimizerAt V E) |
| (hR : ∀ i : Fin 3,E=ᵐ[torusVolume]((axisReflection i).torus ⁻¹' E)) |
| {b : Fin 3 →Bool} |
| (hd : ∀ x∈upperOctant,∀ y∈upperOctant,OrientedCoordinateLE b x y → |
| quotientMap y∈openRepresentative E →quotientMap x∈openRepresentative E) |
| (i : Fin 3) (j : Fin 2) (t : ℝ) |
| (hin : quotientMap (axialSectionCorner b i t (fun k ↦ if k=j then false else true))∈openRepresentative E) |
| (hout : quotientMap (axialSectionCorner b i t (fun k ↦ if k=j then true else false))∉closure (openRepresentative E)) |
| {e r : ℝ} (he : 0≤e) (hr : r∈Icc 0 (21/100 : ℝ)) |
| (hedge : ∀ y∈closure (centeredAxialSlice (openRepresentative E) b i (t : AxisCircle)), |
| (∀ k : Fin 2,k≠j →y k=((1/2 : ℝ) : AxisCircle)) →2*‖y j‖≤e) |
| (hgap : r≤volume.real (axialSlice (openRepresentative E) i (t : AxisCircle))-e) : |
| ENNReal.ofReal (2*(1+(27/20)*r^2))≤planarTorusPerimeter (axialSlice (openRepresentative E) i (t : AxisCircle)) := by |
| let A := centeredAxialSlice (openRepresentative E) b i (t : AxisCircle) |
| have hA : IsOpen A := centeredAxialSlice_isOpen (isOpen_openRepresentative E) b i _ |
| have hdA : PlanarRadialDownset A := isMinimizerAt_centered_section_radial hV hmin hR hd i _ |
| have hg : r≤volume.real A-e := by simpa only [A,centeredAxialSlice_volume] using hgap |
| suffices hp : ENNReal.ofReal (2*(1+(27/20)*r^2))≤planarTorusPerimeter A by |
| simpa only [A,centeredAxialSlice_perimeter] using hp |
| fin_cases j |
| · have hi : planarAxisPoint (1/2)∈A := by |
| change (torusAxialSplit i).symm ((t : AxisCircle),planarAxisPoint (1/2)+axialSectionCenter b i)∈openRepresentative E |
| rw [centered_section_second_axis] |
| convert hin using 2 |
| congr 1 |
| funext k; fin_cases k <;> decide |
| have ho : planarFirstAxisPoint (1/2)∉closure A := by |
| intro hh |
| apply hout |
| have hc := centeredAxialSlice_closure_subset (openRepresentative E) b i (t : AxisCircle) hh |
| change (torusAxialSplit i).symm ((t : AxisCircle),planarFirstAxisPoint (1/2)+axialSectionCenter b i)∈closure (openRepresentative E) at hc |
| rw [centered_section_first_axis] at hc |
| convert hc using 2 |
| congr 1 |
| apply planar_radial_strip_gain_first hA hdA hi ho he hr ?_ hg |
| intro y hy hy1 |
| apply hedge y hy |
| intro k hk |
| fin_cases k |
| · exact (hk rfl).elim |
| · exact hy1 |
| · have hi : planarFirstAxisPoint (1/2)∈A := by |
| change (torusAxialSplit i).symm ((t : AxisCircle),planarFirstAxisPoint (1/2)+axialSectionCenter b i)∈openRepresentative E |
| rw [centered_section_first_axis] |
| convert hin using 2 |
| congr 1 |
| funext k; fin_cases k <;> decide |
| have ho : planarAxisPoint (1/2)∉closure A := by |
| intro hh |
| apply hout |
| have hc := centeredAxialSlice_closure_subset (openRepresentative E) b i (t : AxisCircle) hh |
| change (torusAxialSplit i).symm ((t : AxisCircle),planarAxisPoint (1/2)+axialSectionCenter b i)∈closure (openRepresentative E) at hc |
| rw [centered_section_second_axis] at hc |
| convert hc using 2 |
| congr 1 |
| apply planar_radial_strip_gain_second hA hdA hi ho he hr ?_ hg |
| intro y hy hy0 |
| apply hedge y hy |
| intro k hk |
| fin_cases k |
| · exact hy0 |
| · exact (hk rfl).elim |
|
|
| end CubicTorus |
|
|
| end |
| end |
|
|
| section |
|
|
| noncomputable section |
| open MeasureTheory Set Filter |
| open scoped ENNReal NNReal Topology RealInnerProductSpace |
| namespace CubicTorus |
|
|
| lemma strip_gain_dominates_shape {δ q : ℝ} (_hδ : 0≤δ) (_hq : q∈Icc 0 1) : |
| Real.sqrt (1+Shape.c*δ^2*Shape.shape q)≤1+(27/20)*(δ*q)^2 := by |
| have hshape : Shape.shape q≤q^2 := min_le_left _ _ |
| have hmul := mul_le_mul_of_nonneg_left hshape (show 0≤(27/10 : ℝ)*δ^2 by positivity) |
| apply (Real.sqrt_le_iff).mpr |
| refine ⟨by positivity,?_⟩ |
| dsimp only [Shape.c] at * |
| nlinarith [sq_nonneg ((27/20 : ℝ)*(δ*q)^2)] |
|
|
| lemma disk_gain_dominates_shape {δ q s : ℝ} (hδ : δ∈Icc 0 Shape.dm) |
| (hq : q∈Icc 0 1) (hs : (38/100 : ℝ)<s) : |
| Real.sqrt (1+Shape.c*δ^2*Shape.shape q)≤Real.sqrt (Real.pi*s) := by |
| have hd2 : δ^2≤Shape.dm^2 := by nlinarith [mul_nonneg hδ.1 (sub_nonneg.mpr hδ.2)] |
| have hh : δ^2*Shape.shape q≤Shape.dm^2 := |
| (mul_le_mul_of_nonneg_left (Shape.shape_le_one hq) (sq_nonneg δ)).trans (by simpa using hd2) |
| have hs0 : 0<s := by linarith |
| have hpi := mul_lt_mul_of_pos_right Real.pi_gt_three hs0 |
| apply Real.sqrt_le_sqrt |
| dsimp only [Shape.dm,Shape.c] at * |
| nlinarith |
|
|
| lemma complement_gain_dominates_shape {δ q s u : ℝ} (hδ : δ∈Icc 0 Shape.dm) |
| (hq : q∈Icc 0 1) (hu : u≤1-1/Real.pi) (hs : s=u-δ+δ*q) : |
| Real.sqrt (1+Shape.c*δ^2*Shape.shape q)≤Real.sqrt (Real.pi*(1-s)) := by |
| have hc : 0<Shape.c*Shape.dm := by norm_num [Shape.c,Shape.dm] |
| have hb : Shape.c*Shape.dm*Shape.shape q≤Real.pi*(1-q) := by |
| have hh : Shape.shape q≤Real.pi*(1-q)/(Shape.c*Shape.dm) := min_le_right _ _ |
| have := (le_div_iff₀ hc).mp hh |
| nlinarith |
| have hscale := mul_le_mul_of_nonneg_left hb hδ.1 |
| have hremain := mul_nonneg (mul_nonneg hδ.1 (sub_nonneg.mpr hδ.2)) (Shape.shape_nonneg hq) |
| have hu' := mul_le_mul_of_nonneg_left hu Real.pi_pos.le |
| have hp : Real.pi*(1/Real.pi)=1 := by field_simp |
| rw [hs] |
| apply Real.sqrt_le_sqrt |
| dsimp only [Shape.c,Shape.dm] at * |
| nlinarith |
|
|
| end CubicTorus |
|
|
| end |
| end |
|
|
| section |
|
|
| noncomputable section |
| open MeasureTheory Set Filter |
| open scoped ENNReal NNReal Topology RealInnerProductSpace |
| namespace CubicTorus |
|
|
| theorem isMinimizerAt_section_improved_profile {V : ℝ} (hV : V∈Ioo 0 1) |
| {E : Set Torus} (hmin : IsMinimizerAt V E) |
| (hR : ∀ i : Fin 3,E=ᵐ[torusVolume]((axisReflection i).torus ⁻¹' E)) |
| {b : Fin 3 →Bool} |
| (hd : ∀ x∈upperOctant,∀ y∈upperOctant,OrientedCoordinateLE b x y → |
| quotientMap y∈openRepresentative E →quotientMap x∈openRepresentative E) |
| (i : Fin 3) {t : ℝ} (_ht : t∈Icc (1/2 : ℝ) 1) |
| (hc : ∀ n : Fin 2 →Bool,quotientMap (axialSectionCorner b i t n)∉frontier (openRepresentative E)) |
| (hn : volume (axialSlice (frontier (openRepresentative E)) i (t : AxisCircle))=0) |
| {e δ q u : ℝ} (he : 0≤e) (hδ : δ∈Icc 0 Shape.dm) (hq : q∈Icc 0 1) |
| (hu : u≤1-1/Real.pi) (heq : e≤u-δ) |
| (hs : volume.real (axialSlice (openRepresentative E) i (t : AxisCircle))=u-δ+δ*q) |
| (hslo : (38/100 : ℝ)<volume.real (axialSlice (openRepresentative E) i (t : AxisCircle))) |
| (hedge : ∀ j : Fin 2,∀ y∈closure (centeredAxialSlice (openRepresentative E) b i (t : AxisCircle)), |
| (∀ k : Fin 2,k≠j →y k=((1/2 : ℝ) : AxisCircle)) →2*‖y j‖≤e) : |
| ENNReal.ofReal (2*Real.sqrt (1+Shape.c*δ^2*Shape.shape q))≤ |
| planarTorusPerimeter (axialSlice (openRepresentative E) i (t : AxisCircle)) := by |
| have hj : δ*q∈Icc 0 (21/100 : ℝ) := ⟨mul_nonneg hδ.1 hq.1, |
| (mul_le_mul_of_nonneg_left hq.2 hδ.1).trans (by simpa only [mul_one,Shape.dm] using hδ.2)⟩ |
| have hgap : δ*q≤volume.real (axialSlice (openRepresentative E) i (t : AxisCircle))-e := by rw [hs]; linarith |
| have hstrip : 2*Real.sqrt (1+Shape.c*δ^2*Shape.shape q)≤2*(1+(27/20)*(δ*q)^2) := |
| mul_le_mul_of_nonneg_left (strip_gain_dominates_shape hδ.1 hq) (by norm_num) |
| have hdisk : 2*Real.sqrt (1+Shape.c*δ^2*Shape.shape q)≤2*Real.sqrt (Real.pi*volume.real (axialSlice (openRepresentative E) i (t : AxisCircle))) := |
| mul_le_mul_of_nonneg_left (disk_gain_dominates_shape hδ hq hslo) (by norm_num) |
| have hcomp : 2*Real.sqrt (1+Shape.c*δ^2*Shape.shape q)≤2*Real.sqrt (Real.pi*(1-volume.real (axialSlice (openRepresentative E) i (t : AxisCircle)))) := |
| mul_le_mul_of_nonneg_left (complement_gain_dominates_shape hδ hq hu hs) (by norm_num) |
| let A := centeredAxialSlice (openRepresentative E) b i (t : AxisCircle) |
| have hr : PlanarRadialDownset A := isMinimizerAt_centered_section_radial hV hmin hR hd i _ |
| have hv : volume.real A=volume.real (axialSlice (openRepresentative E) i (t : AxisCircle)) := |
| centeredAxialSlice_volume _ b i _ |
| have hcorner (n : Fin 2 →Bool) : planarSquareCorner n∈A ↔ |
| quotientMap (axialSectionCorner b i t n)∈openRepresentative E := by |
| change (torusAxialSplit i).symm ((t : AxisCircle),planarSquareCorner n+axialSectionCenter b i)∈openRepresentative E ↔ _ |
| rw [centered_section_corner] |
| have hout (n : Fin 2 →Bool) (hn : quotientMap (axialSectionCorner b i t n)∉openRepresentative E) : |
| quotientMap (axialSectionCorner b i t n)∉closure (openRepresentative E) := |
| open_outside_closure_of_not_frontier (isOpen_openRepresentative E) (hc n) hn |
| by_cases hnear : quotientMap (axialSectionCorner b i t (fun _ ↦ false))∈openRepresentative E |
| · by_cases hfar : quotientMap (axialSectionCorner b i t (fun _ ↦ true))∈openRepresentative E |
| · have hAu : A=univ := by |
| apply eq_univ_of_forall |
| intro y |
| apply hr _ _ (fun j ↦ ?_) ((hcorner _).mpr hfar) |
| simp only [planarSquareCorner,ite_true] |
| rw [axisCircle_norm_coe_small (by norm_num : |(1/2 : ℝ)|≤1/2)] |
| simpa using axisCircle_norm_le_half (y j) |
| have hs1 : volume.real (axialSlice (openRepresentative E) i (t : AxisCircle))=1 := by |
| rw [←hv,hAu]; simp |
| have hsle : volume.real (axialSlice (openRepresentative E) i (t : AxisCircle))≤u := by |
| rw [hs] |
| nlinarith [mul_le_mul_of_nonneg_left hq.2 hδ.1] |
| have hp : 0<1/Real.pi := one_div_pos.mpr Real.pi_pos |
| exfalso |
| linarith |
| · by_cases h0 : quotientMap (axialSectionCorner b i t (fun j ↦ j==0))∈openRepresentative E |
| · by_cases h1 : quotientMap (axialSectionCorner b i t (fun j ↦ j==1))∈openRepresentative E |
| · obtain ⟨L,hL,hband,hcal,hp⟩ := isMinimizerAt_section_complement_calibration hV hmin hR hd i t (hout _ hfar) h0 h1 hn |
| exact (ENNReal.ofReal_le_ofReal hcomp).trans hp |
| · have he0 : (fun k : Fin 2 ↦ if k=1 then false else true)=(fun k ↦ k==0) := by |
| funext k; fin_cases k <;> decide |
| have he1 : (fun k : Fin 2 ↦ if k=1 then true else false)=(fun k ↦ k==1) := by |
| funext k; fin_cases k <;> decide |
| have hp := isMinimizerAt_section_strip_gain hV hmin hR hd i 1 t (by rwa [he0]) (by rw [he1]; exact hout _ h1) he hj (hedge 1) hgap |
| exact (ENNReal.ofReal_le_ofReal hstrip).trans hp |
| · by_cases h1 : quotientMap (axialSectionCorner b i t (fun j ↦ j==1))∈openRepresentative E |
| · have he0 : (fun k : Fin 2 ↦ if k=0 then false else true)=(fun k ↦ k==1) := by |
| funext k; fin_cases k <;> decide |
| have he1 : (fun k : Fin 2 ↦ if k=0 then true else false)=(fun k ↦ k==0) := by |
| funext k; fin_cases k <;> decide |
| have hp := isMinimizerAt_section_strip_gain hV hmin hR hd i 0 t (by rwa [he0]) (by rw [he1]; exact hout _ h0) he hj (hedge 0) hgap |
| exact (ENNReal.ofReal_le_ofReal hstrip).trans hp |
| · exact (ENNReal.ofReal_le_ofReal hdisk).trans |
| (isMinimizerAt_section_disk_perimeter hV hmin hR hd i t hnear (hout _ h0) (hout _ h1)) |
| · have hAe : A=∅ := by |
| apply eq_empty_iff_forall_notMem.mpr |
| intro y hy |
| apply hnear |
| apply (hcorner _).mp |
| apply hr _ _ _ hy |
| intro j |
| simp only [planarSquareCorner,Bool.false_eq_true,ite_false,norm_zero] |
| exact norm_nonneg _ |
| have hs0 : volume.real (axialSlice (openRepresentative E) i (t : AxisCircle))=0 := by |
| rw [←hv,hAe]; simp |
| exfalso |
| linarith |
|
|
| end CubicTorus |
|
|
| end |
| end |
|
|
| section |
|
|
| noncomputable section |
| open MeasureTheory Set Filter |
| open scoped ENNReal NNReal Topology RealInnerProductSpace |
| namespace CubicTorus |
|
|
| lemma isMinimizerAt_boundary_slice_null_ae {V : ℝ} (hV : V∈Ioo 0 1) |
| {E : Set Torus} (hE : MeasurableSet E) (hmin : IsMinimizerAt V E) (i : Fin 3) : |
| ∀ᵐ t : AxisCircle,volume (axialSlice (frontier (openRepresentative E)) i t)=0 := by |
| apply (lintegral_eq_zero_iff (axialSliceVolume_measurable isClosed_frontier.measurableSet i)).mp |
| rw [axialSliceVolume_lintegral isClosed_frontier.measurableSet i] |
| exact isMinimizerAt_volume_frontier_eq_zero hV hE hmin |
|
|
| theorem isMinimizerAt_planar_profile_ae {V : ℝ} (hV : V∈Ioo 0 1) |
| {E : Set Torus} (hE : MeasurableSet E) (hmin : IsMinimizerAt V E) |
| (hR : ∀ i : Fin 3,E=ᵐ[torusVolume]((axisReflection i).torus ⁻¹' E)) |
| {b : Fin 3 →Bool} |
| (hd : ∀ x∈upperOctant,∀ y∈upperOctant,OrientedCoordinateLE b x y → |
| quotientMap y∈openRepresentative E →quotientMap x∈openRepresentative E) |
| (i : Fin 3) : |
| ∀ᵐ t : ℝ ∂volume.restrict (Icc (1/2 : ℝ) 1), |
| ENNReal.ofReal (unitPlanarProfile (volume.real (axialSlice (openRepresentative E) i (t : AxisCircle))))≤ |
| planarTorusPerimeter (axialSlice (openRepresentative E) i (t : AxisCircle)) := by |
| have hc : ∀ᵐ t : ℝ,∀ n : Fin 2 →Bool, |
| quotientMap (axialSectionCorner b i t n)∉frontier (openRepresentative E) := by |
| rw [ae_all_iff] |
| intro n |
| exact (ae_iff.mpr (by simpa only [not_not] using isMinimizerAt_corner_boundary_heights_null hV hE hmin hR b i n)) |
| have hp : MeasurePreserving (fun t : ℝ ↦ (t : AxisCircle)) (volume.restrict (Ioc (0 : ℝ) 1)) volume := by |
| simpa using AddCircle.measurePreserving_mk (1 : ℝ) 0 |
| have hn := hp.quasiMeasurePreserving.ae (isMinimizerAt_boundary_slice_null_ae hV hE hmin i) |
| have hsub : Icc (1/2 : ℝ) 1⊆Ioc 0 1 := fun t ht ↦ ⟨by linarith [ht.1],ht.2⟩ |
| filter_upwards [ae_restrict_of_ae hc,ae_restrict_of_ae_restrict_of_subset hsub hn, |
| ae_restrict_mem measurableSet_Icc] with t htc htn ht |
| exact isMinimizerAt_section_planar_profile hV hmin hR hd i ht htc htn |
|
|
| theorem isMinimizerAt_nonstandard_planar_profile {V : ℝ} (hV : V∈Ioc 0 (1/2)) |
| {E : Set Torus} (hmin : IsMinimizerAt V E) (hstd : ¬StandardRegionAt V E) |
| (hR : ∀ i : Fin 3,E=ᵐ[torusVolume]((axisReflection i).torus ⁻¹' E)) : |
| ∃ b : Fin 3 →Bool, |
| (∀ x∈upperOctant,∀ y∈upperOctant,OrientedCoordinateLE b x y → |
| quotientMap y∈openRepresentative E →quotientMap x∈openRepresentative E) ∧ |
| ∀ i : Fin 3,∀ᵐ t : ℝ ∂volume.restrict (Icc (1/2 : ℝ) 1), |
| ENNReal.ofReal (unitPlanarProfile (volume.real (axialSlice (openRepresentative E) i (t : AxisCircle))))≤ |
| planarTorusPerimeter (axialSlice (openRepresentative E) i (t : AxisCircle)) := by |
| have hV' : V∈Ioo (0 : ℝ) 1 := ⟨hV.1,by linarith [hV.2]⟩ |
| have hc : IsPreconnected (frontier (openRepresentative E)) := |
| (Periodic.isMinimizerAt_connected_or_standard_slab hV' hmin).resolve_right (fun h ↦ hstd h.2.2) |
| obtain ⟨b,hd⟩ := isMinimizerAt_octant_downset hV' hmin hc hR |
| refine ⟨b,hd,fun i ↦ ?_⟩ |
| let F := toMeasurable torusVolume E |
| have hF : MeasurableSet F := measurableSet_toMeasurable _ _ |
| have hFE : F=ᵐ[torusVolume]E := hmin.1.1.toMeasurable_ae_eq |
| have hmF := (isMinimizerAt_congr_ae hFE).mpr hmin |
| have ho := openRepresentative_congr_ae hFE |
| have hRF (j : Fin 3) : F=ᵐ[torusVolume]((axisReflection j).torus ⁻¹' F) := |
| hFE.trans ((hR j).trans ((axisReflection j).measurePreserving.quasiMeasurePreserving.preimage_ae_eq hFE.symm)) |
| have hdF : ∀ x∈upperOctant,∀ y∈upperOctant,OrientedCoordinateLE b x y → |
| quotientMap y∈openRepresentative F →quotientMap x∈openRepresentative F := by rwa [ho] |
| have hp := isMinimizerAt_planar_profile_ae hV' hF hmF hRF hdF i |
| rwa [ho] at hp |
|
|
| end CubicTorus |
|
|
| end |
| end |
|
|
| section |
|
|
| noncomputable section |
| open MeasureTheory Set Filter |
| open scoped ENNReal NNReal Topology RealInnerProductSpace |
| namespace CubicTorus |
|
|
| lemma octantHeight_surjective_half (b : Fin 3 →Bool) (i : Fin 3) {t : ℝ} |
| (ht : t∈Icc (1/2 : ℝ) 1) : ∃ z∈Icc (0 : ℝ) 1,octantHeight b i z=t := by |
| cases hb : b i |
| · refine ⟨2-2*t,⟨by linarith [ht.2],by linarith [ht.1]⟩,?_⟩ |
| simp only [octantHeight,hb,Bool.false_eq_true,ite_false] |
| ring |
| · refine ⟨2*t-1,⟨by linarith [ht.1],by linarith [ht.2]⟩,?_⟩ |
| simp only [octantHeight,hb,ite_true] |
| ring |
|
|
| theorem isMinimizerAt_nesting_improved_profile_ae {V : ℝ} (hV : V∈Ioo 0 1) |
| {E : Set Torus} (hE : MeasurableSet E) (hmin : IsMinimizerAt V E) |
| (hR : ∀ i : Fin 3,E=ᵐ[torusVolume]((axisReflection i).torus ⁻¹' E)) |
| (b : Fin 3 →Bool) |
| (hd : ∀ x∈upperOctant,∀ y∈upperOctant,OrientedCoordinateLE b x y → |
| quotientMap y∈openRepresentative E →quotientMap x∈openRepresentative E) |
| {μ : FiniteMeasure NormalBundle} (hμ : IsNormalMeasure E μ) {H : ℝ} |
| (hCMC : ∀ F : Cover → Cover,ContDiff ℝ (⊤ : ℕ∞) F → |
| (∀ m x,F (x+latticeVector m)=F x) → |
| (∫ z,tangentialDivergence F z.1 z.2 ∂(μ : Measure NormalBundle))= |
| H*(∫ z,⟪(z.2 : Cover),F z.1⟫ ∂(μ : Measure NormalBundle))) |
| (i : Fin 3) (ha : (axialAreaMarginal μ i : Measure AxisCircle)≪volume) |
| (hv : quotientMap (orientedUpperVertex b (fun j ↦ if j=i then true else false))∉openRepresentative E) |
| (hout : quotientMap (axialSectionCorner b i (octantHeight b i 0) (fun _ ↦ true))∉closure (openRepresentative E)) |
| (hin0 : quotientMap (axialSectionCorner b i (octantHeight b i 0) (fun j ↦ j==0))∈openRepresentative E) |
| (hin1 : quotientMap (axialSectionCorner b i (octantHeight b i 0) (fun j ↦ j==1))∈openRepresentative E) |
| (hH : H/2∈Icc (51/50 : ℝ) (81/50 : ℝ)) : |
| let s := volume.real (axialSlice (openRepresentative E) i (octantHeight b i 0 : AxisCircle)) |
| let δ := nestingDelta (H/2) s |
| 0<δ →∀ᵐ t : ℝ ∂volume.restrict (Icc (1/2 : ℝ) 1), |
| let v := volume.real (axialSlice (openRepresentative E) i (t : AxisCircle)) |
| v∈Icc (Shape.u (H/2)-δ) (Shape.u (H/2)) → |
| ENNReal.ofReal (2*Real.sqrt (1+Shape.c*δ^2*Shape.shape ((v-Shape.u (H/2)+δ)/δ)))≤ |
| planarTorusPerimeter (axialSlice (openRepresentative E) i (t : AxisCircle)) := by |
| intro s δ hδpos |
| have hδ : δ∈Icc 0 Shape.dm := nestingDelta_mem _ _ |
| have hh : 0<H/2 := by linarith [hH.1] |
| obtain ⟨_,hdisc,_⟩ := isMinimizerAt_bottom_both_extents hV hE hmin hR b hd hμ hCMC i ha hv hout hin0 hin1 |
| have hbound := nestingEndpointBound_ge_exact hh (show 0 ≤ s from measureReal_nonneg) hdisc |
| let e := max 0 (nestingEndpointBound (H/2) s) |
| have he : 0≤e := le_max_left _ _ |
| have hegap : e≤Shape.u (H/2)-δ := nestingDelta_gap hH hδpos |
| have hplace := Shape.interval_placement hH |
| have hc : ∀ᵐ t : ℝ,∀ n : Fin 2 →Bool, |
| quotientMap (axialSectionCorner b i t n)∉frontier (openRepresentative E) := by |
| rw [ae_all_iff] |
| intro n |
| exact ae_iff.mpr (by simpa only [not_not] using isMinimizerAt_corner_boundary_heights_null hV hE hmin hR b i n) |
| have hp : MeasurePreserving (fun t : ℝ ↦ (t : AxisCircle)) (volume.restrict (Ioc (0 : ℝ) 1)) volume := by |
| simpa using AddCircle.measurePreserving_mk (1 : ℝ) 0 |
| have hn := hp.quasiMeasurePreserving.ae (isMinimizerAt_boundary_slice_null_ae hV hE hmin i) |
| have hsub : Icc (1/2 : ℝ) 1⊆Ioc 0 1 := fun t ht ↦ ⟨by linarith [ht.1],ht.2⟩ |
| filter_upwards [ae_restrict_of_ae hc,ae_restrict_of_ae_restrict_of_subset hsub hn, |
| ae_restrict_mem measurableSet_Icc] with t htc htn ht |
| intro v hvrange |
| let q := (v-Shape.u (H/2)+δ)/δ |
| have hq : q∈Icc (0 : ℝ) 1 := ⟨div_nonneg (by linarith [hvrange.1]) hδpos.le, |
| (div_le_one hδpos).mpr (by linarith [hvrange.2])⟩ |
| have hvrepr : v=Shape.u (H/2)-δ+δ*q := by dsimp only [q]; field_simp; ring |
| have hvlo : (38/100 : ℝ)<v := by linarith [hplace.2.1,hδ.2,hvrange.1] |
| apply isMinimizerAt_section_improved_profile hV hmin hR hd i ht htc htn he hδ hq hplace.2.2.2.1 hegap hvrepr hvlo |
| intro j y hy hyedge |
| obtain ⟨z,hz,hzt⟩ := octantHeight_surjective_half b i ht |
| have hb := isMinimizerAt_bottom_nested_endpoint hV hE hmin hR b hd hμ hCMC i ha hv hout hin0 hin1 j z hz y (by rwa [hzt]) hyedge |
| exact (hb.trans hbound).trans (le_max_right _ _) |
|
|
| end CubicTorus |
|
|
| end |
| end |
|
|
| section |
|
|
| noncomputable section |
| open MeasureTheory Set Filter |
| open scoped ENNReal NNReal Topology RealInnerProductSpace |
| namespace CubicTorus |
|
|
| lemma axialSliceIntegral_measurable {E : Set Torus} (hE : MeasurableSet E) (i : Fin 3) |
| {f : Torus → ℝ} (hf : Measurable f) : Measurable (axialSliceIntegral E i f) := by |
| have hh := ((hf.indicator hE).comp (torusAxialSplit i).symm.measurable).stronglyMeasurable.integral_prod_right' (ν := (volume : Measure TransverseProduct)) |
| simpa only [Function.comp_apply,←axialSliceIntegral_indicator hE] using hh.measurable |
|
|
| lemma axialTestPerimeter_countable (i : Fin 3) : |
| ∃ S : Set C(Torus,ℝ),S⊆axialDivergenceClass i ∧ S.Countable ∧ |
| ∀ (E : Set Torus) (t : AxisCircle),axialTestPerimeter E i t= |
| ⨆ f : S,ENNReal.ofReal (axialSliceFunctional E i t f) := by |
| obtain ⟨S,hS,hSc,hSd⟩ := (TopologicalSpace.IsSeparable.of_separableSpace (axialDivergenceClass i)).exists_countable_dense_subset |
| refine ⟨S,hS,hSc,?_⟩ |
| intro E t |
| apply le_antisymm |
| · refine iSup_le fun F ↦ iSup_le fun hF ↦ iSup_le fun hFi ↦ ?_ |
| let f : C(Torus,ℝ) := ⟨fun z ↦ divergence F (representative z), |
| continuous_periodic_descent (continuous_divergence hF.1) (divergence_periodic hF.2.1)⟩ |
| have hf : f∈axialDivergenceClass i := ⟨F,hF,hFi,rfl⟩ |
| let r : ℝ≥0∞ := ⨆ f : S,ENNReal.ofReal (axialSliceFunctional E i t f) |
| have hc : IsClosed {f : C(Torus,ℝ) | ENNReal.ofReal (axialSliceFunctional E i t f)≤r} := |
| isClosed_le (ENNReal.continuous_ofReal.comp (axialSliceFunctional E i t).continuous) continuous_const |
| have hh : S⊆{f : C(Torus,ℝ) | ENNReal.ofReal (axialSliceFunctional E i t f)≤r} := by |
| intro f hf |
| exact le_iSup (fun f : S ↦ ENNReal.ofReal (axialSliceFunctional E i t f)) ⟨f,hf⟩ |
| exact (closure_minimal hh hc) (hSd hf) |
| · apply iSup_le |
| intro f |
| obtain ⟨F,hF,hFi,hfe⟩ := hS f.property |
| have he : axialSliceFunctional E i t f= |
| axialSliceIntegral E i (fun z ↦ divergence F (representative z)) t := by |
| change axialSliceIntegral E i f t=_ |
| rw [hfe] |
| rfl |
| rw [he] |
| exact le_iSup_of_le F (le_iSup_of_le hF (le_iSup_of_le hFi le_rfl)) |
|
|
| theorem planarTorusPerimeter_slice_measurable {E : Set Torus} (hE : MeasurableSet E) (i : Fin 3) : |
| Measurable (fun t : AxisCircle ↦ planarTorusPerimeter (axialSlice E i t)) := by |
| obtain ⟨S,hS,hSc,he⟩ := axialTestPerimeter_countable i |
| have : Countable S := hSc.to_subtype |
| have hh : Measurable (fun t : AxisCircle ↦ ⨆ f : S,ENNReal.ofReal (axialSliceFunctional E i t f)) := by |
| apply Measurable.iSup |
| intro f |
| exact (axialSliceIntegral_measurable hE i f.1.continuous.measurable).ennreal_ofReal |
| simpa only [←he,axialTestPerimeter_eq_planar] using hh |
|
|
| end CubicTorus |
|
|
| end |
| end |
|
|
| section |
|
|
| noncomputable section |
| open MeasureTheory Set Filter |
| open scoped ENNReal NNReal Topology RealInnerProductSpace |
| namespace CubicTorus |
|
|
| lemma axisReflection_axialSplit (i : Fin 3) (t : AxisCircle) (y : TransverseProduct) : |
| (axisReflection i).torus ((torusAxialSplit i).symm (t,y))= |
| (torusAxialSplit i).symm (-t,y) := by |
| ext j |
| rw [axisReflection_torus_apply] |
| by_cases hj : j=i |
| · subst j; simp |
| · obtain ⟨k,rfl⟩ := Fin.exists_succAbove_eq hj |
| simp [Fin.succAbove_ne] |
|
|
| lemma axialSlice_neg_of_reflection {O : Set Torus} |
| (hR : ∀ i : Fin 3,∀ z : Torus,(axisReflection i).torus z∈O ↔ z∈O) |
| (i : Fin 3) (t : AxisCircle) : axialSlice O i (-t)=axialSlice O i t := by |
| ext y |
| change (torusAxialSplit i).symm (-t,y)∈O ↔ (torusAxialSplit i).symm (t,y)∈O |
| rw [←axisReflection_axialSplit] |
| exact hR i _ |
|
|
| end CubicTorus |
| end |
| end |
| end |
|
|
| end OAI |
|
|
