| import OAI.Geometry.ProjectionBodies.SphericalCore |
|
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| namespace OAI |
|
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| universe uE |
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| noncomputable section |
| open Set MeasureTheory Metric Filter Topology Function |
| open scoped ENNReal NNReal RealInnerProductSpace Pointwise |
| namespace PettyProjection |
|
|
| /-- The actual support function of a compact body, not a surrogate gauge. -/ |
| def support {n : ℕ} (K : Set (Space n)) (u : Space n) : ℝ := |
| sSup ((fun x => ⟪u,x⟫) '' K) |
|
|
| lemma support_attained {n : ℕ} {K : Set (Space n)} (hK : IsCompact K) (hKn : K.Nonempty) |
| (u : Space n) : ∃ x ∈ K, support K u = ⟪u,x⟫ := |
| hK.exists_sSup_image_eq hKn (by fun_prop) |
|
|
| lemma inner_le_support {n : ℕ} {K : Set (Space n)} (hK : IsCompact K) |
| {x : Space n} (hx : x ∈ K) (u : Space n) : ⟪u,x⟫ ≤ support K u := by |
| exact le_csSup (hK.bddAbove_image (by fun_prop)) (mem_image_of_mem _ hx) |
|
|
| lemma support_le_iff {n : ℕ} {K : Set (Space n)} (hK : IsCompact K) (hKn : K.Nonempty) |
| {u : Space n} {a : ℝ} : support K u ≤ a ↔ ∀ x ∈ K, ⟪u,x⟫ ≤ a := by |
| constructor |
| · intro h x hx; exact (inner_le_support hK hx u).trans h |
| · intro h |
| obtain ⟨x,hx,he⟩ := support_attained hK hKn u |
| rw [he]; exact h x hx |
|
|
| lemma support_continuous {n : ℕ} {K : Set (Space n)} (hK : IsCompact K) : |
| Continuous (support K) := |
| hK.continuous_sSup (by fun_prop) |
|
|
| lemma support_lipschitz {n : ℕ} {K : Set (Space n)} (hK : IsCompact K) (hKn : K.Nonempty) |
| {R : ℝ≥0} (hR : ∀ x ∈ K, ‖x‖ ≤ R) : LipschitzWith R (support K) := by |
| apply LipschitzWith.of_le_add_mul |
| intro u v |
| obtain ⟨x,hx,he⟩ := support_attained hK hKn u |
| rw [he] |
| have hi := inner_le_support hK hx v |
| have hbound : ⟪u-v,x⟫ ≤ R*dist u v := by |
| calc |
| _ ≤ ‖u-v‖*‖x‖ := real_inner_le_norm _ _ |
| _ ≤ ‖u-v‖*R := mul_le_mul_of_nonneg_left (hR x hx) (norm_nonneg _) |
| _ = _ := by rw [dist_eq_norm]; ring |
| rw [inner_sub_left] at hbound |
| linarith |
|
|
| lemma support_smul {n : ℕ} {K : Set (Space n)} (hK : IsCompact K) (hKn : K.Nonempty) |
| {r : ℝ} (hr : 0 ≤ r) (u : Space n) : support K (r • u) = r*support K u := by |
| apply le_antisymm |
| · apply (support_le_iff hK hKn).mpr |
| intro x hx |
| rw [real_inner_smul_left] |
| exact mul_le_mul_of_nonneg_left (inner_le_support hK hx u) hr |
| · obtain ⟨x,hx,he⟩ := support_attained hK hKn u |
| rw [he] |
| simpa only [real_inner_smul_left] using inner_le_support hK hx (r • u) |
|
|
| lemma support_add_le {n : ℕ} {K : Set (Space n)} (hK : IsCompact K) (hKn : K.Nonempty) |
| (u v : Space n) : support K (u+v) ≤ support K u+support K v := by |
| apply (support_le_iff hK hKn).mpr |
| intro x hx |
| rw [inner_add_left] |
| exact add_le_add (inner_le_support hK hx u) (inner_le_support hK hx v) |
|
|
| lemma support_nonneg {n : ℕ} {K : Set (Space n)} (hK : IsCompact K) (h0 : 0 ∈ K) |
| (u : Space n) : 0 ≤ support K u := by |
| simpa only [inner_zero_right] using inner_le_support hK h0 u |
|
|
| lemma support_minkowski {n : ℕ} {K L : Set (Space n)} |
| (hK : IsCompact K) (hKn : K.Nonempty) (hL : IsCompact L) (hLn : L.Nonempty) |
| (u : Space n) : support (K+L) u = support K u+support L u := by |
| apply le_antisymm |
| · apply (support_le_iff (hK.add hL) (hKn.add hLn)).mpr |
| rintro _ ⟨x,hx,y,hy,rfl⟩ |
| rw [inner_add_right] |
| exact add_le_add (inner_le_support hK hx u) (inner_le_support hL hy u) |
| · obtain ⟨x,hx,hxe⟩ := support_attained hK hKn u |
| obtain ⟨y,hy,hye⟩ := support_attained hL hLn u |
| rw [hxe,hye,← inner_add_right] |
| exact inner_le_support (hK.add hL) (add_mem_add hx hy) u |
|
|
| lemma support_linear_image {n : ℕ} {K : Set (Space n)} (hK : IsCompact K) (hKn : K.Nonempty) |
| (T : Space n →L[ℝ] Space n) (u : Space n) : |
| support (T '' K) u = support K (T.adjoint u) := by |
| apply le_antisymm |
| · apply (support_le_iff (hK.image T.continuous) (hKn.image T)).mpr |
| rintro _ ⟨x,hx,rfl⟩ |
| rw [← ContinuousLinearMap.adjoint_inner_left] |
| exact inner_le_support hK hx _ |
| · obtain ⟨x,hx,he⟩ := support_attained hK hKn (T.adjoint u) |
| rw [he,ContinuousLinearMap.adjoint_inner_left] |
| exact inner_le_support (hK.image T.continuous) (mem_image_of_mem T hx) u |
|
|
| lemma mem_iff_inner_le_support {n : ℕ} {K : Set (Space n)} |
| (hK : IsCompact K) (hc : Convex ℝ K) (hKn : K.Nonempty) {x : Space n} : |
| x ∈ K ↔ ∀ u : Space n, ⟪u,x⟫ ≤ support K u := by |
| constructor |
| · intro hx u; exact inner_le_support hK hx u |
| · intro hx |
| by_contra hn |
| obtain ⟨f,a,hfa,hax⟩ := geometric_hahn_banach_closed_point hc hK.isClosed hn |
| let u := (InnerProductSpace.toDual ℝ (Space n)).symm f |
| have hu (y : Space n) : ⟪u,y⟫ = f y := by |
| change (InnerProductSpace.toDual ℝ (Space n)) u y = f y |
| rw [LinearIsometryEquiv.apply_symm_apply] |
| obtain ⟨y,hy,he⟩ := support_attained hK hKn u |
| have h := hx u |
| rw [he,hu,hu] at h |
| exact (not_lt_of_ge h) ((hfa y hy).trans hax) |
|
|
| lemma support_eq_iff {n : ℕ} {K L : Set (Space n)} |
| (hK : IsCompact K) (hcK : Convex ℝ K) (hKn : K.Nonempty) |
| (hL : IsCompact L) (hcL : Convex ℝ L) (hLn : L.Nonempty) : |
| support K = support L ↔ K = L := by |
| constructor |
| · intro h |
| ext x |
| rw [mem_iff_inner_le_support hK hcK hKn,mem_iff_inner_le_support hL hcL hLn,h] |
| · rintro rfl; rfl |
|
|
| end PettyProjection |
| end |
|
|
| noncomputable section |
| open Set MeasureTheory Metric Filter Topology |
| open scoped ENNReal RealInnerProductSpace Pointwise |
| namespace PettyProjection.Spherical |
|
|
| lemma harmonicSpace_isHilbertSum {n : ℕ} (hn : 2 ≤ n) [NeZero n] : |
| IsHilbertSum ℝ (fun d : ℕ => harmonicSpace n d) |
| (fun d : ℕ => (harmonicSpace n d).subtypeₗᵢ) := by |
| let F : ℕ → Submodule ℝ (H n) := fun d => harmonicSpace n d |
| have horth : OrthogonalFamily ℝ (fun d => F d) (fun d => (F d).subtypeₗᵢ) := |
| harmonicSpace_orthogonal hn |
| have htotal : (⨆ d, F d).topologicalClosure = ⊤ := harmonicSpace_total hn |
| let : ∀ d, CompleteSpace (F d) := fun d => FiniteDimensional.complete ℝ (harmonicSpace n d) |
| exact IsHilbertSum.mkInternal F horth htotal.ge |
|
|
| open MvPolynomial |
|
|
| lemma polynomialL2_X_mul_X (n : ℕ) [NeZero n] (i j : Fin n) : |
| polynomialL2 n (X i * X j) = toH n (coordinateProduct n i j) := by |
| change toH n (polynomialRestriction n (X i * X j)) = _ |
| congr 1 |
| ext u |
| simp [polynomialRestriction_apply, polynomialFunction, coordinateProduct] |
|
|
| lemma polynomialL2_X_mul_mem_lowSpace {n : ℕ} [NeZero n] (i : Fin n) |
| {p : MvPolynomial (Fin n) ℝ} (hp : p.IsHomogeneous 1) : |
| polynomialL2 n (X i * p) ∈ lowSpace n := by |
| have hmem : p ∈ Submodule.span ℝ (Set.range (X : Fin n → MvPolynomial (Fin n) ℝ)) := by |
| rw [← homogeneousSubmodule_one_eq_span_X] |
| exact hp |
| clear hp |
| induction hmem using Submodule.span_induction with |
| | mem q hq => |
| obtain ⟨j, rfl⟩ := hq |
| rw [polynomialL2_X_mul_X] |
| exact Submodule.subset_span ⟨(i,j), rfl⟩ |
| | zero => simp |
| | add p q _ _ hp hq => |
| simpa only [mul_add, map_add] using Submodule.add_mem (lowSpace n) hp hq |
| | smul c p _ hp => |
| simpa only [mul_smul_comm, map_smul] using Submodule.smul_mem (lowSpace n) c hp |
|
|
| lemma homogeneous_two_mem_lowSpace {n : ℕ} [NeZero n] {p : MvPolynomial (Fin n) ℝ} |
| (hp : p.IsHomogeneous 2) : polynomialL2 n p ∈ lowSpace n := by |
| have hsum : polynomialL2 n (∑ i : Fin n, X i * pderiv i p) ∈ lowSpace n := by |
| rw [map_sum] |
| exact Submodule.sum_mem _ fun i _ => polynomialL2_X_mul_mem_lowSpace i hp.pderiv |
| have htwo : (2 : ℝ) • polynomialL2 n p ∈ lowSpace n := by |
| simpa only [hp.sum_X_mul_pderiv, two_nsmul, map_add, two_smul ℝ] using hsum |
| exact (Submodule.smul_mem_iff (lowSpace n) (by norm_num : (2 : ℝ) ≠ 0)).mp htwo |
|
|
| lemma harmonicSpace_two_le_lowSpace (n : ℕ) [NeZero n] : |
| harmonicSpace n 2 ≤ lowSpace n := by |
| rintro v ⟨p, hp, rfl⟩ |
| exact homogeneous_two_mem_lowSpace hp.1 |
|
|
| end PettyProjection.Spherical |
| end |
|
|
| noncomputable section |
| open MeasureTheory Set Filter Topology |
| open scoped ENNReal RealInnerProductSpace |
| namespace PettyProjection |
|
|
| variable {E : Type uE} [NormedAddCommGroup E] [InnerProductSpace ℝ E] [CompleteSpace E] |
| variable {F : ℕ → Submodule ℝ E} [∀ i, CompleteSpace (F i)] |
|
|
| lemma hilbertSum_component (h : IsHilbertSum ℝ (fun i => F i) (fun i => (F i).subtypeₗᵢ)) |
| (x : E) (i : ℕ) : |
| (F i).starProjection x = (h.linearIsometryEquiv x i : E) := by |
| apply Submodule.eq_starProjection_of_mem_of_inner_eq_zero (h.linearIsometryEquiv x i).property |
| intro w hw |
| have hh := h.linearIsometryEquiv.symm.inner_map_map (h.linearIsometryEquiv x) |
| (lp.single 2 i (⟨w,hw⟩ : F i)) |
| rw [LinearIsometryEquiv.symm_apply_apply, h.linearIsometryEquiv_symm_apply_single, |
| lp.inner_single_right] at hh |
| change ⟪x, w⟫ = ⟪(h.linearIsometryEquiv x i : E), w⟫ at hh |
| rw [inner_sub_left, hh, sub_self] |
|
|
| lemma hilbertSum_hasSum_projection (h : IsHilbertSum ℝ (fun i => F i) (fun i => (F i).subtypeₗᵢ)) |
| (x : E) : HasSum (fun i => (F i).starProjection x) x := by |
| have hh := h.hasSum_linearIsometryEquiv_symm (h.linearIsometryEquiv x) |
| rw [LinearIsometryEquiv.symm_apply_apply] at hh |
| simpa only [hilbertSum_component h, Submodule.coe_subtypeₗᵢ, Submodule.coe_subtype] using hh |
|
|
| lemma hilbertSum_hasSum_projection_norm_sq |
| (h : IsHilbertSum ℝ (fun i => F i) (fun i => (F i).subtypeₗᵢ)) (x : E) : |
| HasSum (fun i => ‖(F i).starProjection x‖ ^ 2) (‖x‖ ^ 2) := by |
| have hh := lp.hasSum_norm (by norm_num : 0 < (2 : ℝ≥0∞).toReal) (h.linearIsometryEquiv x) |
| simpa only [ENNReal.toReal_ofNat, Real.rpow_two, LinearIsometryEquiv.norm_map, |
| hilbertSum_component h, Submodule.norm_coe] using hh |
|
|
| end PettyProjection |
| end |
|
|
| end OAI |
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|
