| import OAI.Geometry.ProjectionBodies.Homothety |
|
|
| namespace OAI |
|
|
| universe uι |
|
|
| noncomputable section |
| open Set MeasureTheory Metric Filter Topology Function |
| open scoped ENNReal NNReal RealInnerProductSpace Pointwise Gradient |
| namespace PettyProjection |
| open Spherical (Sphere norm_coe sigma mean) |
|
|
| lemma gauge_pos_sphere {n : ℕ} [NeZero n] {K : Set (Space n)} |
| (hK : IsCompact K) (hc : Convex ℝ K) (h0 : K ∈ 𝓝 (0 : Space n)) |
| (u : Sphere n) : 0 < gauge K (u : Space n) := by |
| let s : C(Sphere n,ℝ) := ⟨fun u => support K u,(support_continuous hK).comp continuous_subtype_val⟩ |
| have hs : ∀ u,0 < s u := support_pos_of_nhds hK h0 |
| have hw : wulff s = K := wulff_support hK hc ⟨0,mem_of_mem_nhds h0⟩ |
| rw [← hw,gauge_wulff_eq_max s hs] |
| exact wulffMax_pos s hs (by intro h; simpa [h] using norm_coe u) |
|
|
| def gaugeDensity {n : ℕ} (K : Set (Space n)) (u : Sphere n) : ENNReal := |
| ENNReal.ofReal ((gauge K (u : Space n))⁻¹^n) |
|
|
| def gaugeProjectionMeasure {n : ℕ} (K : Set (Space n)) : Measure (Space n) := |
| Measure.map (fun u : Sphere n => ∇ (gauge K) (u : Space n)) |
| ((sigma n).withDensity (gaugeDensity K)) |
|
|
| lemma gaugeDensity_measurable {n : ℕ} {K : Set (Space n)} |
| (hc : Convex ℝ K) (h0 : K ∈ 𝓝 (0 : Space n)) : Measurable (gaugeDensity K) := |
| ENNReal.measurable_ofReal.comp ((((continuous_gauge hc h0).measurable.comp |
| measurable_subtype_coe).inv).pow_const n) |
|
|
| lemma gaugeDensity_toReal {n : ℕ} (K : Set (Space n)) (u : Sphere n) : |
| (gaugeDensity K u).toReal=(gauge K (u : Space n))⁻¹^n := by |
| exact ENNReal.toReal_ofReal (pow_nonneg (inv_nonneg.mpr (gauge_nonneg _)) _) |
|
|
| lemma gaugeMeasure_integral {n : ℕ} {K : Set (Space n)} |
| (hc : Convex ℝ K) (h0 : K ∈ 𝓝 (0 : Space n)) {f : Space n → ℝ} |
| (hf : StronglyMeasurable f) : |
| ∫ x,f x ∂gaugeProjectionMeasure K = |
| mean (fun u : Sphere n => (gauge K (u : Space n))⁻¹^n*f (∇ (gauge K) (u : Space n))) := by |
| have hm : Measurable (fun u : Sphere n => ∇ (gauge K) (u : Space n)) := |
| (measurable_gradient _).comp measurable_subtype_coe |
| rw [gaugeProjectionMeasure,integral_map_of_stronglyMeasurable hm hf, |
| integral_withDensity_eq_integral_toReal_smul (gaugeDensity_measurable hc h0) |
| (Eventually.of_forall (fun _ => ENNReal.ofReal_lt_top))] |
| simp only [gaugeDensity_toReal,smul_eq_mul,mean] |
|
|
| lemma gaugeMeasure_finite {n : ℕ} [NeZero n] {K : Set (Space n)} |
| (hK : IsCompact K) (hc : Convex ℝ K) (h0 : K ∈ 𝓝 (0 : Space n)) : |
| IsFiniteMeasure (gaugeProjectionMeasure K) := by |
| have hp := gauge_pos_sphere hK hc h0 |
| have hcont : Continuous (fun u : Sphere n => (gauge K (u : Space n))⁻¹^n) := |
| (((continuous_gauge hc h0).comp continuous_subtype_val).inv₀ (fun u => (hp u).ne')).pow n |
| have hi := hcont.integrable_of_hasCompactSupport (μ := sigma n) (HasCompactSupport.of_compactSpace _) |
| have hf : ∫⁻ u, gaugeDensity K u ∂sigma n ≠ ⊤ := by |
| unfold gaugeDensity |
| rw [← ofReal_integral_eq_lintegral_ofReal hi |
| (Eventually.of_forall (fun u => (pow_nonneg (inv_nonneg.mpr (hp u).le) n)))] |
| exact ENNReal.ofReal_ne_top |
| have := isFiniteMeasure_withDensity hf |
| exact Measure.isFiniteMeasure_map _ _ |
|
|
| lemma gaugeMeasure_mass {n : ℕ} [NeZero n] {K : Set (Space n)} |
| (hK : IsCompact K) (hc : Convex ℝ K) (h0 : K ∈ 𝓝 (0 : Space n)) : |
| (gaugeProjectionMeasure K).real univ=volume.real K/kappa n := by |
| have hi := gaugeMeasure_integral hc h0 (f := fun _ => 1) stronglyMeasurable_const |
| simp only [integral_const,smul_eq_mul,mul_one] at hi |
| rw [hi,gauge_volume hc h0 (gauge_pos_sphere hK hc h0)] |
| field_simp [(kappa_pos n).ne'] |
|
|
| lemma gaugeMeasure_probability {n : ℕ} [NeZero n] {K : Set (Space n)} |
| (hK : IsCompact K) (hc : Convex ℝ K) (h0 : K ∈ 𝓝 (0 : Space n)) |
| (hv : volume.real K=kappa n) : IsProbabilityMeasure (gaugeProjectionMeasure K) := by |
| constructor |
| apply (ENNReal.toReal_eq_one_iff _).mp |
| rw [show (gaugeProjectionMeasure K univ).toReal=(gaugeProjectionMeasure K).real univ from rfl, |
| gaugeMeasure_mass hK hc h0,hv,div_self (kappa_pos n).ne'] |
|
|
| lemma gaugeMeasure_bounded {n : ℕ} {K : Set (Space n)} |
| (hc : Convex ℝ K) (h0 : K ∈ 𝓝 (0 : Space n)) : |
| ∃ B : ℝ, 0 ≤ B ∧ ∀ᵐ x ∂gaugeProjectionMeasure K, ‖x‖ ≤ B := by |
| obtain ⟨C,hC⟩ := hc.lipschitz_gauge h0 |
| refine ⟨C,C.coe_nonneg,?_⟩ |
| have hm : Measurable (fun u : Sphere n => ∇ (gauge K) (u : Space n)) := |
| (measurable_gradient _).comp measurable_subtype_coe |
| apply (ae_map_iff hm.aemeasurable |
| (measurableSet_le measurable_norm measurable_const)).mpr |
| apply Eventually.of_forall |
| intro u |
| have he : ‖∇ (gauge K) (u : Space n)‖=‖fderiv ℝ (gauge K) (u : Space n)‖ := |
| (InnerProductSpace.toDual ℝ (Space n)).symm.norm_map _ |
| rw [he] |
| exact norm_fderiv_le_of_lipschitz ℝ hC |
|
|
| lemma boundedMeasure_integrable_continuous {n : ℕ} {μ : Measure (Space n)} [IsFiniteMeasure μ] |
| (hb : ∃ B : ℝ, 0 ≤ B ∧ ∀ᵐ x ∂μ, ‖x‖ ≤ B) {f : Space n → ℝ} |
| (hf : Continuous f) : Integrable f μ := by |
| obtain ⟨B,_,hB⟩ := hb |
| obtain ⟨C,hC⟩ := (isCompact_closedBall (0 : Space n) B).exists_bound_of_continuousOn hf.continuousOn |
| apply (integrable_const C).mono' hf.aestronglyMeasurable |
| filter_upwards [hB] with x hx |
| exact hC x (by simpa only [mem_closedBall,dist_zero_right] using hx) |
|
|
| lemma gaugeMeasure_shadow {n : ℕ} [NeZero n] {K : Set (Space n)} |
| (hK : IsCompact K) (hc : Convex ℝ K) (h0 : K ∈ 𝓝 (0 : Space n)) (u : Sphere n) : |
| 2*shadowVolume K u=n*kappa n*∫ x,|⟪x,(u : Space n)⟫| ∂gaugeProjectionMeasure K := by |
| rw [gaugeMeasure_integral hc h0 (by fun_prop)] |
| exact shadow_gauge_formula hK hc h0 u |
|
|
| lemma gaugeMeasure_firstVariation {n : ℕ} [NeZero n] {K M : Set (Space n)} |
| (hK : IsCompact K) (hcK : Convex ℝ K) (h0 : K ∈ 𝓝 (0 : Space n)) |
| (hM : IsCompact M) (hcM : Convex ℝ M) (hMn : M.Nonempty) : |
| HasDerivWithinAt (fun t : ℝ => volume.real (K+t • M)) |
| (n*kappa n*∫ x,support M x ∂gaugeProjectionMeasure K) (Ici 0) 0 := by |
| rw [gaugeMeasure_integral hcK h0 (support_continuous hM).stronglyMeasurable] |
| exact minkowski_volume_derivative hK hcK h0 hM hcM hMn |
|
|
| end PettyProjection |
| end |
|
|
| noncomputable section |
| open Set MeasureTheory Metric Filter Topology Function |
| open scoped ENNReal NNReal RealInnerProductSpace Pointwise Gradient |
| namespace PettyProjection |
| open Spherical (Sphere norm_coe sigma mean) |
|
|
| def supportFunctional {n : ℕ} (μ : Measure (Space n)) (K : Set (Space n)) : ℝ := |
| ∫ x,support K x ∂μ |
|
|
| def normalizedVolume {n : ℕ} (K : Set (Space n)) : ℝ := |
| volumeRoot K/(kappa n)^((n:ℝ)⁻¹) |
|
|
| lemma normalizedVolume_formula {n : ℕ} (K : Set (Space n)) : |
| normalizedVolume K=(volume.real K/kappa n)^((n:ℝ)⁻¹) := by |
| rw [Real.div_rpow measureReal_nonneg (kappa_pos n).le] |
| rfl |
|
|
| lemma gaugeMeasure_functional {n : ℕ} (hn : 2 ≤ n) [NeZero n] |
| {K M : Set (Space n)} (hK : IsCompact K) (hcK : Convex ℝ K) |
| (h0 : K ∈ 𝓝 (0 : Space n)) (hv : volume.real K=kappa n) |
| (hM : IsCompact M) (hcM : Convex ℝ M) (hiM : (interior M).Nonempty) : |
| normalizedVolume M ≤ supportFunctional (gaugeProjectionMeasure K) M ∧ |
| (supportFunctional (gaugeProjectionMeasure K) M=normalizedVolume M → |
| ∃ z : Space n, ∃ c : ℝ, 0 < c ∧ K=(fun x => z+c • x) '' M) := by |
| have hiK : (interior K).Nonempty := ⟨0,mem_interior_iff_mem_nhds.mpr h0⟩ |
| have hV := gaugeMeasure_firstVariation hK hcK h0 hM hcM (hiM.mono interior_subset) |
| obtain ⟨hlow,heq⟩ := minkowski_derivative_bound_equality hn hK hcK hiK hM hcM hiM hV |
| have hnR : 0 < (n:ℝ) := by exact_mod_cast (show 0<n by omega) |
| have hp : 0 < (kappa n)^((n:ℝ)⁻¹) := Real.rpow_pos_of_pos (kappa_pos n) _ |
| have hroot : volumeRoot K=(kappa n)^((n:ℝ)⁻¹) := by unfold volumeRoot; rw [hv] |
| have hmul : n*kappa n*normalizedVolume M=(n:ℝ)*volumeRoot K^(n-1)*volumeRoot M := by |
| have hpow := volumeRoot_pow (show 0<n by omega) K |
| have hsplit : volumeRoot K^(n-1)*volumeRoot K=kappa n := by |
| rw [← pow_succ,Nat.sub_add_cancel (by omega),hpow,hv] |
| rw [normalizedVolume,← hroot,← hsplit] |
| field_simp [(volumeRoot_pos hK hiK).ne'] |
| refine ⟨?_,fun h => heq ?_⟩ |
| · rw [← hmul] at hlow |
| exact (mul_le_mul_iff_right₀ (mul_pos hnR (kappa_pos n))).mp hlow |
| · change n*kappa n*supportFunctional (gaugeProjectionMeasure K) M=_ |
| rw [h,hmul] |
|
|
| lemma gaugeMeasure_self {n : ℕ} [NeZero n] {K : Set (Space n)} |
| (hK : IsCompact K) (hc : Convex ℝ K) (h0 : K ∈ 𝓝 (0 : Space n)) : |
| supportFunctional (gaugeProjectionMeasure K) K=volume.real K/kappa n := by |
| have hKn : K.Nonempty := ⟨0,mem_of_mem_nhds h0⟩ |
| have hV := gaugeMeasure_firstVariation hK hc h0 hK hc hKn |
| have hd : HasDerivAt (fun t : ℝ => (1+t)^n*volume.real K) (n*volume.real K) 0 := by |
| simpa only [id_eq,Pi.pow_apply,add_zero,one_pow,mul_one] using |
| (((hasDerivAt_id (0:ℝ)).const_add 1).pow n).mul_const (volume.real K) |
| have he (t : ℝ) (ht : t ∈ Ici (0:ℝ)) : |
| volume.real (K+t • K)=(1+t)^n*volume.real K := by |
| have he : K+t • K=(1+t) • K := by simpa only [one_smul] using (hc.add_smul (by norm_num : (0:ℝ)≤1) ht).symm |
| rw [he,volumeReal_smul_nonneg (by have := ht; change 0 ≤ t at this; linarith : 0≤1+t)] |
| simp only [finrank_euclideanSpace_fin] |
| have hd' := hd.hasDerivWithinAt (s := Ici (0:ℝ)) |>.congr he (he 0 (by norm_num)) |
| have hdEq := (hV.derivWithin (uniqueDiffWithinAt_Ici (0:ℝ))).symm.trans |
| (hd'.derivWithin (uniqueDiffWithinAt_Ici (0:ℝ))) |
| change n*kappa n*supportFunctional (gaugeProjectionMeasure K) K=n*volume.real K at hdEq |
| have hn : (n:ℝ) ≠ 0 := by exact_mod_cast (NeZero.ne n) |
| apply (eq_div_iff (kappa_pos n).ne').mpr |
| rw [mul_assoc] at hdEq |
| simpa only [mul_comm] using (mul_left_cancel₀ hn hdEq) |
|
|
| lemma shadowVolume_pos {n : ℕ} {K : Set (Space n)} |
| (hK : IsCompact K) (h0 : K ∈ 𝓝 (0 : Space n)) (u : Space n) : 0 < shadowVolume K u := by |
| obtain ⟨r,hr,hrK⟩ := Metric.mem_nhds_iff.mp h0 |
| have hball : ball (0 : perpendicular u) r ⊆ (perpendicular u).orthogonalProjectionOnto '' K := by |
| intro z hz |
| refine ⟨z,hrK ?_,?_⟩ |
| · simpa only [mem_ball,dist_zero_right,Submodule.norm_coe] using hz |
| · exact Submodule.orthogonalProjectionOnto_mem_subspace_eq_self z |
| have hpos : 0 < (volume : Measure (perpendicular u)) |
| ((perpendicular u).orthogonalProjectionOnto '' K) := |
| (measure_ball_pos volume 0 hr).trans_le (measure_mono hball) |
| exact ENNReal.toReal_pos hpos.ne' |
| (hK.image (perpendicular u).orthogonalProjectionOnto.continuous).measure_ne_top |
|
|
| end PettyProjection |
| end |
|
|
| noncomputable section |
| open Set MeasureTheory Metric Filter Topology Function |
| open scoped NNReal ENNReal RealInnerProductSpace Gradient |
| namespace PettyProjection.Spherical |
|
|
| lemma convex_deriv_limit {ι : Type uι} {l : Filter ι} {f : ι → ℝ → ℝ} |
| {g : ℝ → ℝ} {d : ι → ℝ} {a : ℝ} |
| (hf : ∀ j, ConvexOn ℝ univ (f j)) |
| (hd : ∀ j, HasDerivAt (f j) (d j) 0) |
| (hg : HasDerivAt g a 0) |
| (hlim : ∀ t, Tendsto (fun j => f j t) l (𝓝 (g t))) : |
| Tendsto d l (𝓝 a) := by |
| have hs (t : ℝ) : Tendsto (fun j => slope (f j) 0 t) l (𝓝 (slope g 0 t)) := by |
| simpa only [slope, vsub_eq_sub, smul_eq_mul] using |
| ((hlim t).sub (hlim 0)).const_mul (t - 0)⁻¹ |
| rw [tendsto_order] |
| constructor |
| · intro b hb |
| have hh : ∀ᶠ t in 𝓝[<] (0 : ℝ), b < slope g 0 t := |
| (hg.tendsto_slope.mono_left (nhdsWithin_mono _ (by grind : Iio (0 : ℝ) ⊆ {0}ᶜ))).eventually |
| (eventually_gt_nhds hb) |
| have ht : ∀ᶠ t in 𝓝[<] (0 : ℝ), t < 0 := self_mem_nhdsWithin |
| obtain ⟨t, ht, hbt⟩ := (ht.and hh).exists |
| filter_upwards [(hs t).eventually (eventually_gt_nhds hbt)] with j hj |
| have hj' := (hf j).slope_le_of_hasDerivAt (mem_univ t) (mem_univ 0) ht (hd j) |
| rw [slope_comm] at hj' |
| exact hj.trans_le hj' |
| · intro b hb |
| have hh : ∀ᶠ t in 𝓝[>] (0 : ℝ), slope g 0 t < b := |
| (hg.tendsto_slope.mono_left (nhdsWithin_mono _ (by grind : Ioi (0 : ℝ) ⊆ {0}ᶜ))).eventually |
| (eventually_lt_nhds hb) |
| have ht : ∀ᶠ t in 𝓝[>] (0 : ℝ), 0 < t := self_mem_nhdsWithin |
| obtain ⟨t, ht, hbt⟩ := (ht.and hh).exists |
| filter_upwards [(hs t).eventually (eventually_lt_nhds hbt)] with j hj |
| exact ((hf j).le_slope_of_hasDerivAt (mem_univ 0) (mem_univ t) ht (hd j)).trans_lt hj |
|
|
| lemma convex_fderiv_apply_limit {n : ℕ} {ι : Type uι} {l : Filter ι} |
| {f : ι → Space n → ℝ} {g : Space n → ℝ} {x : Space n} |
| (hf : ∀ j, ConvexOn ℝ univ (f j)) (hd : ∀ j, DifferentiableAt ℝ (f j) x) |
| (hg : DifferentiableAt ℝ g x) |
| (hlim : ∀ y, Tendsto (fun j => f j y) l (𝓝 (g y))) (w : Space n) : |
| Tendsto (fun j => fderiv ℝ (f j) x w) l (𝓝 (fderiv ℝ g x w)) := by |
| apply convex_deriv_limit (fun j => line_convex (hf j) x w) |
| · intro j |
| have h : HasFDerivAt (f j) (fderiv ℝ (f j) x) (x + (0 : ℝ) • w) := by |
| simpa only [zero_smul, add_zero] using (hd j).hasFDerivAt |
| exact h.comp_hasDerivAt 0 (line_hasDerivAt x w 0) |
| · have h : HasFDerivAt g (fderiv ℝ g x) (x + (0 : ℝ) • w) := by |
| simpa only [zero_smul, add_zero] using hg.hasFDerivAt |
| exact h.comp_hasDerivAt 0 (line_hasDerivAt x w 0) |
| · intro t |
| exact hlim (x + t • w) |
|
|
| lemma convex_gradient_limit {n : ℕ} {ι : Type uι} {l : Filter ι} |
| {f : ι → Space n → ℝ} {g : Space n → ℝ} {x : Space n} |
| (hf : ∀ j, ConvexOn ℝ univ (f j)) (hd : ∀ j, DifferentiableAt ℝ (f j) x) |
| (hg : DifferentiableAt ℝ g x) |
| (hlim : ∀ y, Tendsto (fun j => f j y) l (𝓝 (g y))) : |
| Tendsto (fun j => ∇ (f j) x) l (𝓝 (∇ g x)) := by |
| have hp : Tendsto (fun j i => (∇ (f j) x) i) l (𝓝 (fun i => (∇ g x) i)) := by |
| apply tendsto_pi_nhds.mpr |
| intro i |
| have h := convex_fderiv_apply_limit hf hd hg hlim (EuclideanSpace.single i 1) |
| simpa only [← inner_gradient_left, EuclideanSpace.inner_single_right, one_mul, conj_trivial] using h |
| exact (PiLp.continuous_toLp 2 (fun _ : Fin n => ℝ)).continuousAt.tendsto.comp hp |
|
|
| lemma normSmooth_tendsto {n : ℕ} (g : Seminorm ℝ (Space n)) (x : Space n) : |
| Tendsto (fun j => normSmooth g (shrinkingBump n j) x) atTop (𝓝 (g x)) := by |
| obtain ⟨C, hC⟩ := seminorm_lipschitz n g |
| apply tendsto_iff_dist_tendsto_zero.mpr |
| exact squeeze_zero (fun _ => dist_nonneg) (fun j => normSmooth_dist_le g _ hC x) |
| (by simpa only [mul_zero, shrinkingBump] using |
| (tendsto_one_div_add_atTop_nhds_zero_nat (𝕜 := ℝ)).const_mul (C : ℝ)) |
|
|
| lemma normSmooth_gradient_tendsto {n : ℕ} (g : Seminorm ℝ (Space n)) {x : Space n} |
| (hx : DifferentiableAt ℝ (g : Space n → ℝ) x) : |
| Tendsto (fun j => ∇ (normSmooth g (shrinkingBump n j)) x) atTop (𝓝 (∇ (g : Space n → ℝ) x)) := |
| convex_gradient_limit (fun j => normSmooth_convex g _) |
| (fun j => (normSmooth_contDiff g _).differentiable (by norm_num) x) hx (normSmooth_tendsto g) |
|
|
| end PettyProjection.Spherical |
| end |
|
|
| end OAI |
|
|
