import OAI.Geometry.Perimeter.Regularity.SmulMemPhaseSupport namespace OAI noncomputable section open Set Filter MeasureTheory open scoped Topology ENNReal NNReal ContDiff open scoped Topology ENNReal NNReal open scoped ENNReal Topology ContDiff Pointwise open scoped Topology ENNReal ContDiff open scoped BoundedContinuousFunction open scoped ENNReal Topology open scoped ENNReal Topology ContDiff open scoped Topology ENNReal namespace RPISingularHarnack def coneRadialCap {n : ℕ} (R : ℝ) (x : Euc n) : ℝ := expNegInvGlue (R ^ 2 - ‖x‖ ^ 2) theorem coneRadialCap_contDiff {n : ℕ} (R : ℝ) : ContDiff ℝ ∞ (coneRadialCap (n := n) R) := expNegInvGlue.contDiff.comp (contDiff_const.sub (contDiff_id.norm_sq ℝ)) theorem coneRadialCap_tsupport {n : ℕ} {R : ℝ} (hR : 0 < R) : tsupport (coneRadialCap (n := n) R) ⊆ Metric.closedBall (0 : Euc n) R := by apply closure_minimal _ Metric.isClosed_closedBall intro x hx have hh : 0 < R ^ 2 - ‖x‖ ^ 2 := by by_contra hh exact hx (expNegInvGlue.zero_of_nonpos (le_of_not_gt hh)) rw [Metric.mem_closedBall, dist_zero_right] nlinarith [norm_nonneg x] theorem gradient_coneRadialCap {n : ℕ} (R : ℝ) (x : Euc n) : gradient (coneRadialCap R) x = (-2 * ((R ^ 2 - ‖x‖ ^ 2)⁻¹ ^ 2 * expNegInvGlue (R ^ 2 - ‖x‖ ^ 2))) • x := by have hg (s : ℝ) : HasDerivAt expNegInvGlue (s⁻¹ ^ 2 * expNegInvGlue s) s := by simpa using expNegInvGlue.hasDerivAt_polynomial_eval_inv_mul 1 s have hq := ((hasFDerivAt_id (𝕜 := ℝ) x).norm_sq).const_sub (R ^ 2) have hh := (hg (R ^ 2 - ‖x‖ ^ 2)).comp_hasFDerivAt x hq change HasFDerivAt (coneRadialCap R) _ x at hh apply (InnerProductSpace.toDual ℝ (Euc n)).injective rw [toDual_gradient, hh.fderiv] ext y simp only [smul_apply, neg_apply, ContinuousLinearMap.comp_apply, ContinuousLinearMap.id_apply, id_eq, innerSL_apply_apply, map_smul, InnerProductSpace.toDual_apply_apply, smul_eq_mul] ring theorem minimizing_cone_support_in_halfspace_plane {n : ℕ} (hn : 1 < n) {E : Set (Euc n)} (hmin : IsLocallyPerimeterMinimizing E) (hcone : IsCone E) (v : Euc n) (hside : ∀ x ∈ perimeterSupport E, 0 ≤ inner ℝ x v) : ∀ x ∈ perimeterSupport E, inner ℝ x v = 0 := by have hs : ∀ᵐ x ∂perimeterMeasure E, x ∈ perimeterSupport E := by rw [← perimeterMeasure_support] exact (perimeterMeasure E).support_mem_ae have h0 : ∀ᵐ x ∂perimeterMeasure E, 0 ≤ inner ℝ x v := hs.mono fun x hx ↦ hside x hx have hball (R : ℝ) (hR : 0 < R) : ∀ᵐ x ∂(perimeterMeasure E).restrict (Metric.ball 0 R), inner ℝ x v = 0 := by let φ : Euc n → ℝ := coneRadialCap R let b : Euc n → ℝ := fun x ↦ (R ^ 2 - ‖x‖ ^ 2)⁻¹ ^ 2 * expNegInvGlue (R ^ 2 - ‖x‖ ^ 2) let V : Euc n → Euc n := fun x ↦ φ x • v have hφ : ContDiff ℝ ∞ φ := coneRadialCap_contDiff R have htφ : HasCompactSupport φ := (isCompact_closedBall (0 : Euc n) R).of_isClosed_subset (isClosed_tsupport _) (coneRadialCap_tsupport hR) have hV : ContDiff ℝ ∞ V := hφ.smul contDiff_const have htV : HasCompactSupport V := htφ.smul_right have hVs : tsupport V ⊆ Metric.ball (0 : Euc n) (R + 1) := (tsupport_smul_subset_left _ _).trans ((coneRadialCap_tsupport hR).trans (Metric.closedBall_subset_ball (by linarith))) have hdiv : ∀ᵐ x ∂perimeterMeasure E, tangentialDivergence E V x = -2 * (b x * inner ℝ x v) := by filter_upwards [minimizing_cone_radial_tangent_ae hn hmin hcone] with x hx rw [tangentialDivergence_smul E (hφ.differentiable (by simp)) (differentiable_const v)] have hc : tangentialDivergence E (fun _ : Euc n ↦ v) x = 0 := by simp [tangentialDivergence, divergence] rw [hc, mul_zero, zero_add, tangentialGradient, gradient_coneRadialCap] simp only [real_inner_smul_left, hx, mul_zero, zero_smul, sub_zero] dsimp [b] ring have hdf : Integrable (fun x ↦ b x * inner ℝ x v) ((perimeterMeasure E).restrict (Metric.ball 0 (R + 1))) := by have hh := (tangentialDivergence_integrable_ball_perimeter hmin 0 (by linarith : 0 < R + 1) hV).const_mul (-1/2 : ℝ) apply hh.congr filter_upwards [ae_restrict_of_ae hdiv] with x hx rw [hx] ring have hnn : ∀ᵐ x ∂(perimeterMeasure E).restrict (Metric.ball 0 (R + 1)), 0 ≤ b x * inner ℝ x v := by filter_upwards [ae_restrict_of_ae h0] with x hx exact mul_nonneg (mul_nonneg (sq_nonneg _) (expNegInvGlue.nonneg _)) hx have hint : (∫ x in Metric.ball (0 : Euc n) (R + 1), b x * inner ℝ x v ∂perimeterMeasure E) = 0 := by have hh := minimizing_tangentialDivergence_integral_eq_zero hn hmin 0 (by linarith : 0 < R + 1) hV htV hVs rw [integral_congr_ae (ae_restrict_of_ae hdiv), integral_const_mul] at hh linarith have he := (integral_eq_zero_iff_of_nonneg_ae hnn hdf).mp hint have her : ∀ᵐ x ∂(perimeterMeasure E).restrict (Metric.ball 0 R), b x * inner ℝ x v = 0 := ae_mono (Measure.restrict_mono (Metric.ball_subset_ball (by linarith)) le_rfl) he filter_upwards [her, ae_restrict_mem measurableSet_ball] with x hx hxr have hp : 0 < R ^ 2 - ‖x‖ ^ 2 := by have hh : ‖x‖ < R := by simpa using hxr nlinarith [norm_nonneg x] have hb : 0 < b x := mul_pos (sq_pos_of_pos (inv_pos.mpr hp)) (expNegInvGlue.pos_of_pos hp) exact (mul_eq_zero.mp hx).resolve_left hb.ne' have hae : ∀ᵐ x ∂perimeterMeasure E, inner ℝ x v = 0 := by let B (j : ℕ) := Metric.ball (0 : Euc n) (j + 1 : ℝ) have hcover : (⋃ j, B j) = univ := by apply eq_univ_of_forall intro x obtain ⟨j, hj⟩ := exists_nat_gt ‖x‖ exact mem_iUnion.mpr ⟨j, by simpa [B] using (show ‖x‖ < (j : ℝ) + 1 by linarith)⟩ have hh := (ae_restrict_iUnion_iff B (fun x ↦ inner ℝ x v = 0)).mpr (fun j ↦ hball (j + 1) (by positivity)) simpa only [hcover, Measure.restrict_univ] using hh have hsup := Measure.support_subset_of_isClosed (isClosed_eq (continuous_id.inner continuous_const) continuous_const) hae rw [perimeterMeasure_support] at hsup exact hsup end RPISingularHarnack end noncomputable section open Set Filter MeasureTheory open scoped Topology ENNReal NNReal ContDiff open scoped Topology ENNReal NNReal open scoped ENNReal Topology ContDiff Pointwise open scoped Topology ENNReal ContDiff open scoped BoundedContinuousFunction open scoped ENNReal Topology open scoped ENNReal Topology ContDiff open scoped Topology ENNReal namespace RPISingularHarnack theorem minimizing_cone_containing_halfspace_ae_eq {n : ℕ} (hn : 1 < n) {E : Set (Euc n)} (hmin : IsLocallyPerimeterMinimizing E) (hcone : IsCone E) (hvertex : (0 : Euc n) ∈ perimeterSupport E) {v : Euc n} (hv : ‖v‖ = 1) (hinside : {x | inner ℝ x v < 0} ⊆ E) : E =ᵐ[volume] {x | inner ℝ x v < 0} := by let K := (volume.restrict E).support have hK : IsClosed K := Measure.isClosed_support have hP : IsOpen {x : Euc n | inner ℝ x v < 0} := isOpen_lt (continuous_id.inner continuous_const) continuous_const have hPK : {x : Euc n | inner ℝ x v < 0} ⊆ K := by intro x hx apply Measure.interior_inter_support refine ⟨?_, ?_⟩ · exact (interior_mono hinside) (hP.interior_eq.symm ▸ hx) · rw [Measure.support_eq_univ]; trivial have hPint : {x : Euc n | inner ℝ x v < 0} ⊆ interior K := by exact hP.interior_eq ▸ interior_mono hPK have hfK : frontier K = perimeterSupport E := minimizing_frontier_phaseSupport (Real.HolderConjugate.conjExponent (by exact_mod_cast hn : 1 < (n : ℝ))) hmin have hside : ∀ x ∈ perimeterSupport E, 0 ≤ inner ℝ x v := by intro x hx by_contra hh have hi := hPint (lt_of_not_ge hh) rw [← hfK, frontier] at hx exact hx.2 hi have hplane := minimizing_cone_support_in_halfspace_plane hn hmin hcone v hside have hcover : {x : Euc n | 0 < inner ℝ x v} ⊆ interior K ∪ Kᶜ := by intro x hx by_cases hk : x ∈ K · left by_contra hi have hs : x ∈ perimeterSupport E := by rw [← hfK, frontier, hK.closure_eq] exact ⟨hk, hi⟩ exact (ne_of_gt hx) (hplane x hs) · exact Or.inr hk have hc : IsPreconnected {x : Euc n | 0 < inner ℝ x v} := by have hh := (convex_halfSpace_gt (innerSL ℝ v).toLinearMap.isLinear (0 : ℝ)).isPreconnected change IsPreconnected {x : Euc n | 0 < inner ℝ v x} at hh simpa only [real_inner_comm v] using hh have hor := hc.subset_or_subset isOpen_interior hK.isOpen_compl (disjoint_compl_right.mono_left interior_subset) hcover have hEK : E =ᵐ[volume] K := minimizing_ae_eq_phaseSupport (Real.HolderConjugate.conjExponent (by exact_mod_cast hn : 1 < (n : ℝ))) hmin rcases hor with hall | hnone · have hae : ∀ᵐ x ∂(volume : Measure (Euc n)), x ∈ K := by filter_upwards [ae_inner_ne_value hv 0] with x hx rcases lt_or_gt_of_ne hx with hn | hp · exact hPK hn · exact interior_subset (hall hp) have hKU : K = univ := by have hh := Measure.support_subset_of_isClosed hK hae rw [Measure.support_eq_univ] at hh exact eq_univ_iff_forall.mpr fun x ↦ hh trivial rw [← hfK, hKU, frontier_univ] at hvertex exact False.elim hvertex · apply hEK.trans apply Filter.eventuallyEqSet_iff.mpr filter_upwards [ae_inner_ne_value hv 0] with x hx rcases lt_or_gt_of_ne hx with hn | hp · exact iff_of_true (hPK hn) hn · exact iff_of_false (hnone hp) (not_lt.mpr hp.le) theorem minimizing_cone_containing_halfspace_support_eq {n : ℕ} (hn : 1 < n) {E : Set (Euc n)} (hmin : IsLocallyPerimeterMinimizing E) (hcone : IsCone E) (hvertex : (0 : Euc n) ∈ perimeterSupport E) {v : Euc n} (hv : ‖v‖ = 1) (hinside : {x | inner ℝ x v < 0} ⊆ E) : perimeterSupport E = perimeterSupport {x | inner ℝ x v < 0} := perimeterSupport_eq_of_ae (minimizing_cone_containing_halfspace_ae_eq hn hmin hcone hvertex hv hinside) end RPISingularHarnack end noncomputable section open Set Filter MeasureTheory open scoped Topology ENNReal NNReal ContDiff open scoped Topology ENNReal NNReal open scoped ENNReal Topology ContDiff Pointwise open scoped Topology ENNReal ContDiff open scoped BoundedContinuousFunction open scoped ENNReal Topology open scoped ENNReal Topology ContDiff open scoped Topology ENNReal namespace RPISingularHarnack theorem measure_eq_of_Iio_finite {μ ν : Measure ℝ} (hfin : ∀ a : ℝ, μ (Iio a) ≠ ⊤) (heq : ∀ a : ℝ, μ (Iio a) = ν (Iio a)) : μ = ν := by apply Measure.ext_of_Ico' μ ν · intro a b _ exact ne_top_of_le_ne_top (hfin b) (measure_mono Ico_subset_Iio_self) · intro a b hab rw [← Iio_sdiff_Iio, measure_sdiff (Iio_subset_Iio hab.le) measurableSet_Iio.nullMeasurableSet (hfin a), measure_sdiff (Iio_subset_Iio hab.le) measurableSet_Iio.nullMeasurableSet (by rw [← heq a]; exact hfin a), heq a, heq b] noncomputable def radialScaledMeasure {n : ℕ} (μ : Measure (Euc n)) (k : ℕ) (t : ℝ) : Measure ℝ := ENNReal.ofReal (t^k) • μ.map (fun x ↦ t * ‖x‖) theorem radialScaledMeasure_Iio {n k : ℕ} (μ : Measure (Euc n)) {a t : ℝ} (ht : 0 < t) : radialScaledMeasure μ k t (Iio a) = ENNReal.ofReal (t^k) * μ (Metric.ball 0 (a/t)) := by have hm : Measurable (fun x : Euc n ↦ t * ‖x‖) := by fun_prop rw [radialScaledMeasure, Measure.smul_apply, smul_eq_mul, Measure.map_apply hm measurableSet_Iio] congr 2 ext x simp only [mem_preimage, mem_Iio, Metric.mem_ball, dist_zero_right] rw [lt_div_iff₀ ht, mul_comm] theorem radialScaledMeasure_eq_of_constant_density {n k : ℕ} (μ : Measure (Euc n)) (hfin : ∀ r : ℝ, 0 < r → μ (Metric.ball 0 r) ≠ ⊤) {a : ℝ} (ha : 0 ≤ a) (hdensity : ∀ r : ℝ, 0 < r → μ.real (Metric.ball 0 r) = a * r^k) {s t : ℝ} (hs : 0 < s) (ht : 0 < t) : radialScaledMeasure μ k s = radialScaledMeasure μ k t := by have hval (t : ℝ) (ht : 0 < t) (r : ℝ) : radialScaledMeasure μ k t (Iio r) = ENNReal.ofReal (a * (max r 0)^k) * (if 0 < r then 1 else 0) := by rw [radialScaledMeasure_Iio μ ht] by_cases hr : 0 < r · rw [ite_eq_left hr, mul_one, max_eq_left hr.le] have hrt : 0 < r/t := div_pos hr ht have hp := hdensity (r/t) hrt have he : μ (Metric.ball 0 (r/t)) = ENNReal.ofReal (a * (r/t)^k) := by rw [← hp, Measure.real, ENNReal.ofReal_toReal (hfin _ hrt)] rw [he, ← ENNReal.ofReal_mul' (mul_nonneg ha (pow_nonneg hrt.le _))] congr 1 rw [div_pow] field_simp · rw [ite_eq_right hr, mul_zero, Metric.ball_eq_empty.mpr (div_nonpos_of_nonpos_of_nonneg (le_of_not_gt hr) ht.le), measure_empty, mul_zero] apply measure_eq_of_Iio_finite · intro r rw [hval s hs r] split_ifs <;> finiteness · intro r rw [hval s hs r, hval t ht r] theorem radialDensity_integral_eq_pushforward {n k : ℕ} (μ : Measure (Euc n)) {ψ : ℝ → ℝ} (hψ : Continuous ψ) {t : ℝ} (ht : 0 < t) : (∫ x, radialDensityIntegrand 0 k ψ t x ∂μ) = ∫ s, ψ (s^2) ∂radialScaledMeasure μ k t := by have hm : Measurable (fun x : Euc n ↦ t * ‖x‖) := by fun_prop have hc : Continuous (fun s : ℝ ↦ ψ (s^2)) := by fun_prop rw [radialScaledMeasure, integral_smul_measure, integral_map_of_stronglyMeasurable hm hc.stronglyMeasurable, ENNReal.toReal_ofReal (pow_nonneg ht.le _)] simp only [radialDensityIntegrand, sub_zero, smul_eq_mul, mul_pow, integral_const_mul] theorem radialDensity_integral_ball_eq {n k : ℕ} (μ : Measure (Euc n)) {ψ : ℝ → ℝ} (hψz : ∀ s, 1 ≤ s → ψ s = 0) {M t : ℝ} (hM : 0 < M) (ht : 1/M < t) : (∫ x in Metric.ball (0 : Euc n) M, radialDensityIntegrand 0 k ψ t x ∂μ) = ∫ x, radialDensityIntegrand 0 k ψ t x ∂μ := by apply setIntegral_eq_integral_of_forall_compl_eq_zero intro x hx have ht0 : 0 < t := (one_div_pos.mpr hM).trans ht have htM : 1 < t*M := (div_lt_iff₀ hM).mp ht have hxM : M ≤ ‖x‖ := by simpa only [Metric.mem_ball, dist_zero_right, not_lt] using hx have h1 : 1 ≤ t*‖x‖ := by nlinarith have h2 := pow_le_pow_left₀ (by norm_num : (0:ℝ) ≤ 1) h1 2 simp only [one_pow, mul_pow] at h2 simp only [radialDensityIntegrand, sub_zero, hψz _ h2, mul_zero] theorem minimizing_radialDensityDerivative_eq_zero_of_constant_density {n : ℕ} {E : Set (Euc n)} (hmin : IsLocallyPerimeterMinimizing E) {a : ℝ} (ha : 0 ≤ a) (hdensity : ∀ r : ℝ, 0 < r → (perimeterMeasure E).real (Metric.ball 0 r) = a*r^(n-1)) {M : ℝ} (hM : 0 < M) {ψ : ℝ → ℝ} (hψ : ContDiff ℝ ∞ ψ) (hψz : ∀ s, 1 ≤ s → ψ s = 0) {t : ℝ} (ht : 1/M < t) : (∫ x in Metric.ball 0 M, radialDensityDerivative 0 (n-1) ψ t x ∂perimeterMeasure E) = 0 := by let μ := (perimeterMeasure E).restrict (Metric.ball 0 M) have : IsFiniteMeasure μ := ⟨by rw [Measure.restrict_apply_univ, perimeterMeasure_open E Metric.isOpen_ball] exact hmin.2.1 0 M hM⟩ have hμ : ∀ᵐ x ∂μ, x ∈ Metric.closedBall 0 M := (ae_restrict_mem measurableSet_ball).mono fun _ hx ↦ Metric.ball_subset_closedBall hx have hd := (radialDensity_integral_hasDerivAt 0 M (n-1) hψ μ hμ t).2 have ht0 : 0 < t := (one_div_pos.mpr hM).trans ht have hfin (r : ℝ) (hr : 0 < r) : perimeterMeasure E (Metric.ball 0 r) ≠ ⊤ := by rw [perimeterMeasure_open E Metric.isOpen_ball] exact (hmin.2.1 0 r hr).ne have he (s : ℝ) (hs : 1/M < s) : (∫ x, radialDensityIntegrand 0 (n-1) ψ s x ∂μ) = ∫ x, radialDensityIntegrand 0 (n-1) ψ t x ∂μ := by have hs0 := (one_div_pos.mpr hM).trans hs change (∫ x in Metric.ball 0 M, _ ∂perimeterMeasure E) = ∫ x in Metric.ball 0 M, _ ∂perimeterMeasure E rw [radialDensity_integral_ball_eq _ hψz hM hs, radialDensity_integral_ball_eq _ hψz hM ht, radialDensity_integral_eq_pushforward _ hψ.continuous hs0, radialDensity_integral_eq_pushforward _ hψ.continuous ht0, radialScaledMeasure_eq_of_constant_density _ hfin ha hdensity hs0 ht0] have hdz : HasDerivAt (fun s ↦ ∫ x, radialDensityIntegrand 0 (n-1) ψ s x ∂μ) 0 t := by apply (hasDerivAt_const t (∫ x, radialDensityIntegrand 0 (n-1) ψ t x ∂μ)).congr_of_eventuallyEq filter_upwards [lt_mem_nhds ht] with s hs exact he s hs exact hd.unique hdz theorem minimizing_radial_tangent_of_constant_density {n : ℕ} (hn : 1 < n) {E : Set (Euc n)} (hmin : IsLocallyPerimeterMinimizing E) {a : ℝ} (ha : 0 ≤ a) (hdensity : ∀ r : ℝ, 0 < r → (perimeterMeasure E).real (Metric.ball 0 r) = a*r^(n-1)) : ∀ᵐ x ∂perimeterMeasure E, inner ℝ (approximateNormal E x) x = 0 := by let ψ : ℝ → ℝ := fun s ↦ expNegInvGlue (1-s) have hψ : ContDiff ℝ ∞ ψ := expNegInvGlue.contDiff.comp (contDiff_const.sub contDiff_id) have hψz (s : ℝ) (hs : 1 ≤ s) : ψ s = 0 := expNegInvGlue.zero_of_nonpos (by linarith) have hψd (s : ℝ) : deriv ψ s = -((1-s)⁻¹^2 * expNegInvGlue (1-s)) := by have hg : HasDerivAt expNegInvGlue ((1-s)⁻¹^2 * expNegInvGlue (1-s)) (1-s) := by simpa using expNegInvGlue.hasDerivAt_polynomial_eval_inv_mul 1 (1-s) simpa [ψ, Function.comp_def] using (hg.comp s ((hasDerivAt_id s).const_sub 1)).deriv have hlocal (R : ℝ) (hR : 0 < R) : ∀ᵐ x ∂(perimeterMeasure E).restrict (Metric.ball 0 R), inner ℝ (approximateNormal E x) x = 0 := by let M := 2*R let t := 1/R have hM : 0 < M := by dsimp [M]; positivity have ht0 : 0 < t := one_div_pos.mpr hR have ht : 1/M < t := one_div_lt_one_div_of_lt hR (by dsimp [M]; linarith) have htM : 1/t < M := by dsimp [t, M]; rw [one_div_one_div]; linarith let μ := (perimeterMeasure E).restrict (Metric.ball 0 M) have : IsFiniteMeasure μ := ⟨by rw [Measure.restrict_apply_univ, perimeterMeasure_open E Metric.isOpen_ball] exact hmin.2.1 0 M hM⟩ have hμ : ∀ᵐ x ∂μ, x ∈ Metric.closedBall 0 M := (ae_restrict_mem measurableSet_ball).mono fun _ hx ↦ Metric.ball_subset_closedBall hx let V : Euc n → Euc n := fun x ↦ ψ (t^2*‖x‖^2) • x have hV : ContDiff ℝ ∞ V := (hψ.comp (contDiff_const.mul (contDiff_id.norm_sq (𝕜 := ℝ)))).smul contDiff_id have hp : tsupport (fun x : Euc n ↦ ψ (t^2*‖x‖^2)) ⊆ Metric.closedBall 0 (1/t) := by simpa only [sub_zero] using (radialProfile_tsupport (n := n) 0 hψz ht0) have hs : tsupport V ⊆ Metric.closedBall 0 (1/t) := (tsupport_smul_subset_left (fun x : Euc n ↦ ψ (t^2*‖x‖^2)) (id : Euc n → Euc n)).trans hp have hc : HasCompactSupport V := (isCompact_closedBall (0 : Euc n) (1/t)).of_isClosed_subset (isClosed_tsupport _) hs have hdiv := minimizing_tangentialDivergence_integral_eq_zero hn hmin 0 hM hV hc (hs.trans (Metric.closedBall_subset_ball htM)) have hD0 := minimizing_radialDensityDerivative_eq_zero_of_constant_density hmin ha hdensity hM hψ hψz ht have hDi := (radialDensity_integral_hasDerivAt 0 M (n-1) hψ μ hμ t).1 have hVi := (tangentialDivergence_integrable_ball_perimeter hmin 0 hM hV).const_mul (t^((n-1)-1)) let q : Euc n → ℝ := fun x ↦ t^((n-1)-1)*tangentialDivergence E V x - radialDensityDerivative 0 (n-1) ψ t x have hqi : Integrable q μ := hVi.sub hDi have hqformula : ∀ᵐ x ∂μ, q x = (2*t^((n-1)-1)*t^2*(inner ℝ (approximateNormal E x) x)^2) * ((1-t^2*‖x‖^2)⁻¹^2 * expNegInvGlue (1-t^2*‖x‖^2)) := by filter_upwards [ae_restrict_of_ae (minimizing_halfspaceBlowup_ae hn hmin)] with x hx have hpow : t^(n-1) = t^((n-1)-1)*t := by rw [← pow_succ, Nat.sub_add_cancel (by omega : 1 ≤ n-1)] have hvf := tangentialDivergence_radialProfile (by omega : 0