lean/OAI/Geometry/ProjectionBodies/Normalization.lean

import OAI.Geometry.ProjectionBodies.BilinearEquality

namespace OAI

noncomputable section
open Set MeasureTheory Metric Filter Topology
open scoped NNReal RealInnerProductSpace Pointwise
namespace PettyProjection
variable {n : ℕ}

lemma smul_body {K : Set (Space n)} (hK : IsConvexBody K) {a : ℝ} (ha : a≠0) :
    IsConvexBody (a • K) := by
  refine ⟨hK.1.smul a,hK.2.1.smul a,?_⟩
  rw [interior_smul₀ ha]
  exact hK.2.2.image _

lemma translate_body {K : Set (Space n)} (hK : IsConvexBody K) (a : Space n) :
    IsConvexBody ((fun x => a+x) '' K) := by
  refine ⟨hK.1.image (by fun_prop),hK.2.1.translate a,?_⟩
  change (interior ((Homeomorph.addLeft a) '' K)).Nonempty
  rw [← (Homeomorph.addLeft a).image_interior]
  exact hK.2.2.image _

lemma isEllipsoid_linear_image {K : Set (Space n)} (hK : IsEllipsoid K)
    (T : Space n ≃ₗ[ℝ] Space n) : IsEllipsoid (T '' K) := by
  obtain ⟨a,S,rfl⟩ := hK
  refine ⟨T a,S.trans T,?_⟩
  simp only [image_image]
  congr 1
  funext x
  exact map_add T a (S x)

lemma isEllipsoid_translate {K : Set (Space n)} (hK : IsEllipsoid K) (b : Space n) :
    IsEllipsoid ((fun x => b+x) '' K) := by
  obtain ⟨a,T,rfl⟩ := hK
  refine ⟨b+a,T,?_⟩
  simp only [image_image]
  congr 1
  funext x
  exact (add_assoc b a (T x)).symm

lemma isEllipsoid_smul {K : Set (Space n)} (hK : IsEllipsoid K) {c : ℝ} (hc : c≠0) :
    IsEllipsoid (c • K) := by
  exact isEllipsoid_linear_image hK (LinearEquiv.smulOfNeZero ℝ (Space n) c hc)

lemma ellipsoid_homothety (T : Space n ≃L[ℝ] Space n) (a : Space n) {c : ℝ} (hc : c≠0) :
    IsEllipsoid ((fun x => a+c • x) '' (T '' unitBall n)) := by
  have he : IsEllipsoid (T '' unitBall n) := ⟨0,T.toLinearEquiv,by simp⟩
  have hh := isEllipsoid_translate (isEllipsoid_smul he hc) a
  change IsEllipsoid ((fun x => a+x) '' ((fun x => c • x) '' (T '' unitBall n))) at hh
  simpa only [image_image,Function.comp_def] using hh

def normalizeBody (K : Set (Space n)) : Set (Space n) := (normalizedVolume K)⁻¹ • K

lemma normalizeBody_body {K : Set (Space n)} (hK : IsConvexBody K) : IsConvexBody (normalizeBody K) :=
  smul_body hK (inv_ne_zero (normalizedVolume_pos hK.1 hK.2.2).ne')

lemma normalizeBody_nhds {K : Set (Space n)} (hK : IsCompact K) (h0 : K∈𝓝 (0:Space n)) :
    normalizeBody K∈𝓝 (0:Space n) := by
  apply (set_smul_mem_nhds_zero_iff (inv_ne_zero (normalizedVolume_pos hK ⟨0,mem_interior_iff_mem_nhds.mpr h0⟩).ne')).mpr h0

lemma normalizeBody_volume [NeZero n] {K : Set (Space n)} (hK : IsConvexBody K) : volume.real (normalizeBody K)=kappa n := by
  have hp := normalizedVolume_pos hK.1 hK.2.2
  rw [normalizeBody,volumeReal_smul_nonneg (inv_nonneg.mpr hp.le),finrank_euclideanSpace_fin,inv_pow,normalizedVolume_pow]
  field_simp [(volumeReal_pos_of_body hK.1 hK.2.2).ne',(kappa_pos n).ne']

lemma normalizeBody_ellipsoid {K : Set (Space n)} (hK : IsConvexBody K)
    (he : IsEllipsoid (normalizeBody K)) : IsEllipsoid K := by
  have hh := isEllipsoid_smul he (normalizedVolume_pos hK.1 hK.2.2).ne'
  simpa only [normalizeBody,smul_smul,mul_inv_cancel₀ (normalizedVolume_pos hK.1 hK.2.2).ne',one_smul] using hh

end PettyProjection
end

noncomputable section
open Set MeasureTheory Metric Filter Topology
open scoped NNReal RealInnerProductSpace Pointwise
namespace PettyProjection
variable {n : ℕ}

lemma measurePair_comm {μ ν : Measure (Space n)} [SFinite μ] [SFinite ν]
    (hμ : Integrable (fun x : Space n => ‖x‖) μ)
    (hν : Integrable (fun x : Space n => ‖x‖) ν) : measurePair μ ν=measurePair ν μ := by
  have hi : Integrable (fun z : Space n × Space n => |⟪z.1,z.2⟫|) (μ.prod ν) := by
    apply (hμ.mul_prod hν).mono' (by fun_prop)
    exact Filter.Eventually.of_forall (fun z => by simpa only [Real.norm_eq_abs,abs_abs] using abs_real_inner_le_norm z.1 z.2)
  rw [measurePair,integral_integral_swap hi]
  unfold measurePair
  simp_rw [real_inner_comm]

/-- Main geometric estimate and equality rigidity at the natural volume
normalization; both the input and the projection body remain nonsmooth. -/
theorem normalized_projection {n : ℕ} (hn : 4≤ 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) :
    kappa (n-1)≤ normalizedVolume (projectionBody K) ∧
      (normalizedVolume (projectionBody K)=kappa (n-1) → IsEllipsoid K) := by
  let P := projectionBody K
  let r := normalizedVolume P
  have hP : IsConvexBody P := projectionBody_body hK hc h0
  have hP0 : P∈𝓝 (0:Space n) := projectionBody_nhds hK hc h0
  have hr : 0< r := normalizedVolume_pos hP.1 hP.2.2
  let D := normalizeBody P
  have hD : IsConvexBody D := normalizeBody_body hP
  have hD0 : D∈𝓝 (0:Space n) := normalizeBody_nhds hP.1 hP0
  have hDv : volume.real D=kappa n := normalizeBody_volume hP
  let μ := gaugeProjectionMeasure K
  let ν := gaugeProjectionMeasure D
  let := gaugeMeasure_finite hK hc h0
  let := gaugeMeasure_finite hD.1 hD.2.1 hD0
  have hμ : Integrable (fun x : Space n => ‖x‖) μ := gaugeMeasure_norm_integrable hK hc h0
  have hν : Integrable (fun x : Space n => ‖x‖) ν := gaugeMeasure_norm_integrable hD.1 hD.2.1 hD0
  obtain ⟨hlow,heq⟩ := measure_bilinear hn hμ hν (gaugeMeasure_cosine_pos hK hc h0)
    (gaugeMeasure_cosine_pos hD.1 hD.2.1 hD0)
    (fun M hM => (gaugeMeasure_functional (by omega) hK hc h0 hv hM.1 hM.2.1 hM.2.2).1)
    (fun M hM => (gaugeMeasure_functional (by omega) hD.1 hD.2.1 hD0 hDv hM.1 hM.2.1 hM.2.2).1)
  have hself : supportFunctional ν D=1 := by
    rw [gaugeMeasure_self hD.1 hD.2.1 hD0,hDv,div_self (kappa_pos n).ne']
  have hs : supportFunctional ν D=r⁻¹*((n:ℝ)*kappa n/2)*measurePair μ ν := by
    change (∫ x,support (r⁻¹ • P) x ∂ν)=_
    simp_rw [support_smul_set hP.1 (hP.2.2.mono interior_subset) (inv_nonneg.mpr hr.le)]
    dsimp only [P]
    simp_rw [projectionBody_support hK hc h0]
    rw [integral_const_mul,integral_const_mul]
    change r⁻¹*((n:ℝ)*kappa n/2*(∫ x,cosineSeminorm μ hμ x ∂ν))=_
    rw [← measurePair_cosine hμ,measurePair_comm hν hμ]
    ring
  have hpair : (n:ℝ)*kappa n*measurePair μ ν=2*r := by
    rw [hself] at hs
    have h := congrArg (fun x : ℝ => (2*r)*x) hs
    field_simp at h
    nlinarith [h]
  have hb := ball_cosine_constant (n :=n)
  have hnk : 0<(n:ℝ)*kappa n := mul_pos (Nat.cast_pos.mpr (NeZero.pos n)) (kappa_pos n)
  refine ⟨?_,?_⟩
  · change kappa (n-1)≤ r
    have h := mul_le_mul_of_nonneg_left hlow hnk.le
    nlinarith
  · intro he
    have hp : measurePair μ ν=Spherical.cosineConstant n := by
      change r=kappa (n-1) at he
      apply (mul_left_cancel₀ hnk.ne')
      rw [hpair,he,← hb]
    obtain ⟨T,hT⟩ := heq hp
    have hE : IsConvexBody (T '' unitBall n) := linearEquiv_body T unitBall_body
    obtain ⟨z,c,hcpos,hset⟩ := (gaugeMeasure_functional (by omega) hK hc h0 hv hE.1 hE.2.1 hE.2.2).2 hT
    rw [hset]
    exact ellipsoid_homothety T z hcpos.ne'

end PettyProjection
end

noncomputable section
open Set MeasureTheory Metric Filter Topology
open scoped NNReal RealInnerProductSpace Pointwise
namespace PettyProjection
variable {n : ℕ}

lemma volumeReal_translate (K : Set (Space n)) (a : Space n) :
    volume.real ((fun x => a+x) '' K)=volume.real K := by
  unfold Measure.real
  rw [image_add_left,measure_preimage_add]

lemma normalizedVolume_translate (K : Set (Space n)) (a : Space n) :
    normalizedVolume ((fun x => a+x) '' K)=normalizedVolume K := by
  rw [normalizedVolume_formula,volumeReal_translate,← normalizedVolume_formula]

lemma shadowVolume_translate (K : Set (Space n)) (a u : Space n) :
    shadowVolume ((fun x => a+x) '' K) u=shadowVolume K u := by
  have he : (perpendicular u).orthogonalProjectionOnto '' ((fun x => a+x) '' K)=
      (fun z => (perpendicular u).orthogonalProjectionOnto a+z) ''
        ((perpendicular u).orthogonalProjectionOnto '' K) := by
    simp only [image_image]
    congr 1
    funext x
    exact map_add _ a x
  unfold shadowVolume Measure.real
  rw [he,image_add_left,measure_preimage_add]

lemma projectionBody_translate (K : Set (Space n)) (a : Space n) :
    projectionBody ((fun x => a+x) '' K)=projectionBody K := by
  ext x
  simp only [projectionBody,mem_ofPred_eq,shadowVolume_translate]

lemma projectionRatio_translate (K : Set (Space n)) (a : Space n) :
    projectionRatio ((fun x => a+x) '' K)=projectionRatio K := by
  rw [projectionRatio,projectionBody_translate,volumeReal_translate,projectionRatio]

lemma shadowVolume_smul (K : Set (Space n)) {a : ℝ} (ha : 0≤ a) (u : Spherical.Sphere n) :
    shadowVolume (a • K) u=a^(n-1)*shadowVolume K u := by
  have he : (perpendicular (u:Space n)).orthogonalProjectionOnto '' (a • K)=
      a • ((perpendicular (u:Space n)).orthogonalProjectionOnto '' K) := by
    change _ '' ((fun x : Space n => a • x) '' K)=
      (fun z : perpendicular (u:Space n) => a • z) '' (_ '' K)
    simp only [image_image]
    congr 1
    funext x
    exact map_smul _ a x
  unfold shadowVolume
  rw [he,volumeReal_smul_nonneg ha,perpendicular_finrank]

lemma projectionBody_smul (K : Set (Space n)) {a : ℝ} (ha : 0< a) :
    projectionBody (a • K)=a^(n-1) • projectionBody K := by
  have hp : 0< a^(n-1) := pow_pos ha _
  ext x
  rw [mem_smul_set_iff_inv_smul_mem₀ hp.ne']
  change (∀ u : Space n,‖u‖=1 → ⟪u,x⟫≤ shadowVolume (a • K) u) ↔
    (∀ u : Space n,‖u‖=1 → ⟪u,(a^(n-1))⁻¹ • x⟫≤ shadowVolume K u)
  apply forall_congr'
  intro u
  apply forall_congr'
  intro hu
  have hv := shadowVolume_smul K ha.le (⟨u,by simpa only [mem_sphere,dist_zero_right] using hu⟩ : Spherical.Sphere n)
  change shadowVolume (a • K) u=a^(n-1)*shadowVolume K u at hv
  rw [hv,inner_smul_right]
  rw [show (a^(n-1))⁻¹*⟪u,x⟫=⟪u,x⟫/(a^(n-1)) by ring,div_le_iff₀ hp]
  rw [mul_comm (a^(n-1))]

lemma projectionRatio_smul (K : Set (Space n))
    {a : ℝ} (ha : 0< a) : projectionRatio (a • K)=projectionRatio K := by
  rw [projectionRatio,projectionBody_smul K ha,volumeReal_smul_nonneg (pow_nonneg ha.le _),
    volumeReal_smul_nonneg ha.le,finrank_euclideanSpace_fin,projectionRatio,mul_pow]
  have he : (a^(n-1))^n=(a^n)^(n-1) := by rw [← pow_mul,← pow_mul,Nat.mul_comm]
  rw [he]
  field_simp [ha.ne']

end PettyProjection
end

noncomputable section
open Set MeasureTheory Metric Filter Topology
open scoped NNReal RealInnerProductSpace Pointwise
namespace PettyProjection
variable {n : ℕ}

lemma projectionRatio_normalized {n : ℕ} (hn : 2≤n) [NeZero n] {K : Set (Space n)}
    (hv : volume.real K=kappa n) :
    projectionRatio K=normalizedVolume (projectionBody K)^n*kappa n^(2-(n:ℤ)) := by
  have hn1 : n-1=(n-2)+1 := by omega
  have hn2 : (2-(n:ℤ))=-((n-2:ℕ):ℤ) := by omega
  have hvP : volume.real (projectionBody K)=normalizedVolume (projectionBody K)^n*kappa n := by
    exact (div_eq_iff (kappa_pos n).ne').mp (normalizedVolume_pow (projectionBody K)).symm
  rw [projectionRatio,hv,hvP,hn1,pow_succ,hn2,zpow_neg,zpow_natCast]
  field_simp [(kappa_pos n).ne']

lemma normalized_projection_ratio {n : ℕ} (hn : 4≤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) :
    pettyConstant n≤projectionRatio K ∧ (projectionRatio K=pettyConstant n → IsEllipsoid K) := by
  obtain ⟨hb,he⟩ := normalized_projection hn hK hc h0 hv
  have hkp := kappa_pos (n-1)
  have hpow := zpow_pos (kappa_pos n) (2-(n:ℤ))
  rw [projectionRatio_normalized (by omega) hv]
  unfold pettyConstant
  refine ⟨mul_le_mul_of_nonneg_right (pow_le_pow_left₀ hkp.le hb n) hpow.le,?_⟩
  intro h
  apply he
  apply (pow_left_inj₀ (hkp.le.trans hb) hkp.le (NeZero.ne n)).mp
  exact mul_right_cancel₀ hpow.ne' h

lemma centered_projection_lower_bound {n : ℕ} (hn : 4≤n) [NeZero n] {K : Set (Space n)}
    (hK : IsCompact K) (hc : Convex ℝ K) (h0 : K∈𝓝 (0:Space n)) :
    pettyConstant n≤projectionRatio K ∧ (projectionRatio K=pettyConstant n → IsEllipsoid K) := by
  have hbody : IsConvexBody K := ⟨hK,hc,⟨0,mem_interior_iff_mem_nhds.mpr h0⟩⟩
  have hM := normalizeBody_body hbody
  have hM0 := normalizeBody_nhds hK h0
  obtain ⟨hb,he⟩ := normalized_projection_ratio hn hM.1 hM.2.1 hM0 (normalizeBody_volume hbody)
  have hr : projectionRatio (normalizeBody K)=projectionRatio K :=
    projectionRatio_smul K (inv_pos.mpr (normalizedVolume_pos hK hbody.2.2))
  rw [hr] at hb he
  exact ⟨hb,fun hh => normalizeBody_ellipsoid hbody (he hh)⟩

/-- No origin, symmetry, localness, or boundary smoothness hypothesis is used. -/
theorem projection_lower_bound {n : ℕ} (hn : 4≤n) {K : Set (Space n)}
    (hK : IsConvexBody K) :
    pettyConstant n≤projectionRatio K ∧ (projectionRatio K=pettyConstant n → IsEllipsoid K) := by
  let : NeZero n := ⟨by omega⟩
  obtain ⟨a,ha⟩ := hK.2.2
  let L := (fun x => -a+x) '' K
  have hL : IsConvexBody L := translate_body hK (-a)
  have hL0 : L∈𝓝 (0:Space n) := by
    have hh := (Homeomorph.addLeft (-a)).isOpenMap.image_mem_nhds (mem_interior_iff_mem_nhds.mp ha)
    simpa [L] using hh
  obtain ⟨hb,he⟩ := centered_projection_lower_bound hn hL.1 hL.2.1 hL0
  have hr : projectionRatio L=projectionRatio K := projectionRatio_translate K (-a)
  rw [hr] at hb he
  refine ⟨hb,fun hh => ?_⟩
  have h := isEllipsoid_translate (he hh) a
  simpa only [L,image_image,Function.comp_def,add_neg_cancel_left,image_id'] using h

end PettyProjection
end

end OAI