import OAI.Geometry.LatticeCovering.GaussianSmoothing namespace OAI section noncomputable section noncomputable section open MeasureTheory Filter Set open scoped Topology noncomputable section namespace SingleLatticeCovering.Prekopa open MeasureTheory Set Filter open scoped ENNReal Topology BigOperators lemma integral_RV_zero (f : RV 0 → ℝ) : (∫ x, f x)=f 0 := by rw [Measure.volume_pi_eq_dirac (0 : RV 0), integral_dirac] lemma cons_weighted {d : ℕ} (a b x y : ℝ) (u v : RV d) : Fin.cons (a*x+b*y) (a•u+b•v) = a•Fin.cons x u+b•Fin.cons y v := by ext i refine Fin.cases ?_ (fun j => ?_) i <;> rfl theorem gaussian_dominated_integral (d : ℕ) {f g h : RV d → ℝ} (fc : Continuous f) (gc : Continuous g) (hc : Continuous h) (fp : ∀ x, 0 < f x) (gp : ∀ x, 0 < g x) (hp : ∀ x, 0 < h x) (fb : GaussianBound f) (gb : GaussianBound g) (hb : GaussianBound h) {a b : ℝ} (ha : 0 < a) (hb' : 0 < b) (hab : a+b=1) (hfg : ∀ x y, f x^a*g y^b ≤ h (a•x+b•y)) : (∫ x, f x)^a*(∫ y, g y)^b ≤ ∫ z, h z := by induction d with | zero => simp only [integral_RV_zero] have H := hfg 0 0 simpa only [smul_zero,add_zero] using H | succ d ih => have hm : ∀ y z : RV d, (marginalOne f y)^a*(marginalOne g z)^b ≤ marginalOne h (a•y+b•z) := by intro y z apply positive_integral (fc.comp (show Continuous (fun t : ℝ => Fin.cons t y) by fun_prop)) (gc.comp (show Continuous (fun t : ℝ => Fin.cons t z) by fun_prop)) (fun t => fp (Fin.cons t y)) (fun t => gp (Fin.cons t z)) (fb.integrable_slice fc _) (gb.integrable_slice gc _) (hb.integrable_slice hc _) (fun t => (hp _).le) ha hb' hab intro t u simpa only [cons_weighted,Function.comp_def,id_eq] using hfg (Fin.cons t y) (Fin.cons u z) have H := ih (fb.continuous_marginalOne fc) (gb.continuous_marginalOne gc) (hb.continuous_marginalOne hc) (marginalOne_positive fb fc fp) (marginalOne_positive gb gc gp) (marginalOne_positive hb hc hp) (fb.for_marginalOne fc) (gb.for_marginalOne gc) (hb.for_marginalOne hc) hm simpa only [integral_marginalOne (fb.integrable fc),integral_marginalOne (gb.integrable gc), integral_marginalOne (hb.integrable hc)] using H end SingleLatticeCovering.Prekopa noncomputable section namespace SingleLatticeCovering.Prekopa open MeasureTheory Set Filter Metric open scoped ENNReal Topology BigOperators lemma sum_sq_le_norm_sq (d : ℕ) (x : RV d) : (∑ i, (x i)^2) ≤ (d : ℝ)*‖x‖^2 := by calc _ ≤ ∑ _ : Fin d, ‖x‖^2 := by apply Finset.sum_le_sum intro i _ have hi := norm_le_pi_norm x i simpa only [Real.norm_eq_abs,sq_abs] using pow_le_pow_left₀ (norm_nonneg (x i)) hi 2 _ = _ := by simp def softIndicator {d : ℕ} (K : Set (RV d)) (t : ℝ) (x : RV d) : ℝ := Real.exp (-t*(infDist x K)^2) lemma softIndicator_positive {d : ℕ} (K : Set (RV d)) (t : ℝ) (x : RV d) : 0 < softIndicator K t x := Real.exp_pos _ lemma continuous_softIndicator {d : ℕ} (K : Set (RV d)) (t : ℝ) : Continuous (softIndicator K t) := by unfold softIndicator exact Real.continuous_exp.comp (continuous_const.mul ((continuous_infDist_pt K).pow 2)) lemma softIndicator_gaussianBound {d : ℕ} {K : Set (RV d)} (hK : IsCompact K) (hne : K.Nonempty) {t : ℝ} (ht : 0 < t) : GaussianBound (softIndicator K t) := by obtain ⟨R,hR,hb⟩ := hK.isBounded.exists_pos_norm_le refine ⟨Real.exp (t*R^2),t/(2*((d : ℝ)+1)),Real.exp_pos _,by positivity,?_⟩ intro x obtain ⟨u,hu,he⟩ := hK.exists_infDist_eq_dist hne x have hn : ‖x‖ ≤ infDist x K+R := by calc ‖x‖ ≤ ‖x-u‖+‖u‖ := by simpa only [add_comm] using norm_le_insert' x u _ ≤ infDist x K+R := by rw [he,dist_eq_norm]; exact add_le_add_right (hb u hu) _ have hs : ‖x‖^2 ≤ 2*((infDist x K)^2+R^2) := by have hn' := pow_le_pow_left₀ (norm_nonneg x) hn 2 nlinarith [sq_nonneg (infDist x K-R)] have hs' : (∑ i, (x i)^2) ≤ 2*((d : ℝ)+1)*((infDist x K)^2+R^2) := by calc _ ≤ (d : ℝ)*‖x‖^2 := sum_sq_le_norm_sq d x _ ≤ (d : ℝ)*(2*((infDist x K)^2+R^2)) := mul_le_mul_of_nonneg_left hs (by positivity) _ ≤ _ := by nlinarith [sq_nonneg (infDist x K),sq_nonneg R] rw [abs_of_pos (softIndicator_positive K t x),softIndicator,gaussianKernel,←Real.exp_add] apply Real.exp_le_exp.mpr have hd : 0 < 2*((d : ℝ)+1) := by positivity have hm := mul_le_mul_of_nonneg_left hs' (div_nonneg ht.le hd.le) field_simp at hm ⊢ nlinarith lemma infDist_weighted {d : ℕ} {A B C : Set (RV d)} (hA : IsCompact A) (hB : IsCompact B) (hneA : A.Nonempty) (hneB : B.Nonempty) {a b : ℝ} (ha : 0 ≤ a) (hb : 0 ≤ b) (hABC : ∀ u ∈ A, ∀ v ∈ B, a•u+b•v ∈ C) (x y : RV d) : infDist (a•x+b•y) C ≤ a*infDist x A+b*infDist y B := by obtain ⟨u,hu,heu⟩ := hA.exists_infDist_eq_dist hneA x obtain ⟨v,hv,hev⟩ := hB.exists_infDist_eq_dist hneB y calc _ ≤ dist (a•x+b•y) (a•u+b•v) := infDist_le_dist_of_mem (hABC u hu v hv) _ = ‖a•(x-u)+b•(y-v)‖ := by rw [dist_eq_norm]; congr 1; module _ ≤ ‖a•(x-u)‖+‖b•(y-v)‖ := norm_add_le _ _ _ = _ := by rw [norm_smul,norm_smul,Real.norm_eq_abs,Real.norm_eq_abs, abs_of_nonneg ha,abs_of_nonneg hb,heu,hev,dist_eq_norm,dist_eq_norm] lemma softIndicator_weighted {d : ℕ} {A B C : Set (RV d)} (hA : IsCompact A) (hB : IsCompact B) (hneA : A.Nonempty) (hneB : B.Nonempty) {a b t : ℝ} (ha : 0 ≤ a) (hb : 0 ≤ b) (hab : a+b=1) (ht : 0 ≤ t) (hABC : ∀ u ∈ A, ∀ v ∈ B, a•u+b•v ∈ C) (x y : RV d) : (softIndicator A t x)^a*(softIndicator B t y)^b ≤ softIndicator C t (a•x+b•y) := by have H := infDist_weighted hA hB hneA hneB ha hb hABC x y have hsq : (infDist (a•x+b•y) C)^2 ≤ a*(infDist x A)^2+b*(infDist y B)^2 := by have HH := pow_le_pow_left₀ (infDist_nonneg : 0 ≤ infDist (a•x+b•y) C) H 2 have hm := mul_nonneg (mul_nonneg ha hb) (sq_nonneg (infDist x A-infDist y B)) nlinarith [sq_nonneg (a*infDist x A+b*infDist y B),sq_nonneg (infDist x A),sq_nonneg (infDist y B)] simp only [softIndicator,←Real.exp_mul,←Real.exp_add] apply Real.exp_le_exp.mpr nlinarith [mul_nonneg ht (sub_nonneg.mpr hsq)] end SingleLatticeCovering.Prekopa namespace SingleLatticeCovering.Prekopa open MeasureTheory Set Filter Metric open scoped ENNReal Topology BigOperators lemma softIndicator_tendsto {d : ℕ} {K : Set (RV d)} (hK : IsClosed K) (hne : K.Nonempty) (x : RV d) : Tendsto (fun n : ℕ => softIndicator K ((n : ℝ)+1) x) atTop (𝓝 (K.indicator (fun _ => (1:ℝ)) x)) := by by_cases hx : x ∈ K · simp only [softIndicator,infDist_zero_of_mem hx,zero_pow (by omega : 2 ≠ 0), mul_zero,Real.exp_zero,Set.indicator_of_mem hx] exact tendsto_const_nhds · rw [Set.indicator_of_notMem hx] have hd : 0 < (infDist x K)^2 := sq_pos_of_pos ((hK.notMem_iff_infDist_pos hne).mp hx) have hq : Real.exp (-(infDist x K)^2)<1 := Real.exp_lt_one_iff.mpr (neg_neg_of_pos hd) have H := (tendsto_pow_atTop_nhds_zero_of_lt_one (Real.exp_pos (-(infDist x K)^2)).le hq).mul_const (Real.exp (-(infDist x K)^2)) simp only [zero_mul] at H convert H using 1 ext n rw [softIndicator,←Real.exp_nat_mul,←Real.exp_add] congr 1 ring lemma softIndicator_antitone {d : ℕ} (K : Set (RV d)) (x : RV d) : Antitone (fun t : ℝ => softIndicator K t x) := by intro t u htu apply Real.exp_le_exp.mpr exact mul_le_mul_of_nonneg_right (neg_le_neg htu) (sq_nonneg _) lemma softIndicator_integral_tendsto {d : ℕ} {K : Set (RV d)} (hK : IsCompact K) (hne : K.Nonempty) : Tendsto (fun n : ℕ => ∫ x, softIndicator K ((n : ℝ)+1) x) atTop (𝓝 (volume K).toReal) := by have H := tendsto_integral_of_dominated_convergence (softIndicator K 1) (fun n => (continuous_softIndicator K ((n : ℝ)+1)).aestronglyMeasurable) ((softIndicator_gaussianBound hK hne zero_lt_one).integrable (continuous_softIndicator K 1)) (fun n => Filter.Eventually.of_forall (fun x => by rw [Real.norm_eq_abs,abs_of_pos (softIndicator_positive K _ x)] exact softIndicator_antitone K x (by nlinarith [Nat.cast_nonneg (α := ℝ) n] : (1:ℝ) ≤ (n : ℝ)+1))) (Filter.Eventually.of_forall (softIndicator_tendsto hK.isClosed hne)) simpa only [integral_indicator hK.measurableSet,integral_const,smul_eq_mul,mul_one,measureReal_def,Measure.restrict_apply_univ] using H theorem compact_volume {d : ℕ} {A B C : Set (RV d)} (hA : IsCompact A) (hB : IsCompact B) (hC : IsCompact C) {a b : ℝ} (ha : 0 < a) (hb : 0 < b) (hab : a+b=1) (hABC : ∀ u ∈ A, ∀ v ∈ B, a•u+b•v ∈ C) : ((volume A).toReal)^a*((volume B).toReal)^b ≤ (volume C).toReal := by rcases A.eq_empty_or_nonempty with rfl | hneA · simp only [measure_empty,ENNReal.toReal_zero,Real.zero_rpow (ne_of_gt ha),zero_mul] exact ENNReal.toReal_nonneg rcases B.eq_empty_or_nonempty with rfl | hneB · simp only [measure_empty,ENNReal.toReal_zero,Real.zero_rpow (ne_of_gt hb),mul_zero] exact ENNReal.toReal_nonneg have hneC : C.Nonempty := by obtain ⟨u,hu⟩ := hneA obtain ⟨v,hv⟩ := hneB exact ⟨a•u+b•v,hABC u hu v hv⟩ have hpoint : ∀ n : ℕ, (∫ x, softIndicator A ((n : ℝ)+1) x)^a* (∫ x, softIndicator B ((n : ℝ)+1) x)^b ≤ ∫ x, softIndicator C ((n : ℝ)+1) x := by intro n have hn : 0 < (n : ℝ)+1 := by positivity apply gaussian_dominated_integral d (continuous_softIndicator _ _) (continuous_softIndicator _ _) (continuous_softIndicator _ _) (softIndicator_positive _ _) (softIndicator_positive _ _) (softIndicator_positive _ _) (softIndicator_gaussianBound hA hneA hn) (softIndicator_gaussianBound hB hneB hn) (softIndicator_gaussianBound hC hneC hn) ha hb hab exact softIndicator_weighted hA hB hneA hneB ha.le hb.le hab hn.le hABC exact le_of_tendsto_of_tendsto (((Real.continuous_rpow_const ha.le).tendsto _).comp (softIndicator_integral_tendsto hA hneA) |>.mul (((Real.continuous_rpow_const hb.le).tendsto _).comp (softIndicator_integral_tendsto hB hneB))) (softIndicator_integral_tendsto hC hneC) (Filter.Eventually.of_forall hpoint) end SingleLatticeCovering.Prekopa namespace SingleLatticeCovering.Prekopa open MeasureTheory Set Filter Metric Sections Isotropization open scoped ENNReal Topology BigOperators theorem logConcave_section_volume {m D : ℕ} {K : Set (RV m × E D)} (hK : IsCompact K) (hconv : Convex ℝ K) : LogConcave (fun y => (volume (fiber K y)).toReal) := by refine ⟨fun _ => ENNReal.toReal_nonneg,fun x y a b ha hb hab => ?_⟩ apply compact_volume (fiber_compact hK x) (fiber_compact hK y) (fiber_compact hK (a•x+b•y)) ha hb hab intro u hu v hv have H := hconv hu hv ha.le hb.le hab exact H lemma LogConcave.const_mul {D : ℕ} {f : E D → ℝ} (hf : LogConcave f) {c : ℝ} (hc : 0 ≤ c) : LogConcave (fun x => c*f x) := by refine ⟨fun x => mul_nonneg hc (hf.1 x),fun x y a b ha hb hab => ?_⟩ rcases hc.eq_or_lt with rfl | hc · simp only [zero_mul,Real.zero_rpow (ne_of_gt ha),Real.zero_rpow (ne_of_gt hb),le_refl] · rw [Real.mul_rpow hc.le (hf.1 x),Real.mul_rpow hc.le (hf.1 y)] calc c^a*f x^a*(c^b*f y^b) = (c^a*c^b)*(f x^a*f y^b) := by ring _ = c*(f x^a*f y^b) := by rw [←Real.rpow_add hc,hab,Real.rpow_one] _ ≤ c*f (a•x+b•y) := mul_le_mul_of_nonneg_left (hf.2 x y a b ha hb hab) hc.le theorem logConcave_canonical_density {m D : ℕ} {K : Set (RV m × E D)} (hK : IsCompact K) (hconv : Convex ℝ K) : LogConcave (fun y => ((volume K)⁻¹).toReal*(volume (fiber K y)).toReal) := LogConcave.const_mul (logConcave_section_volume hK hconv) ENNReal.toReal_nonneg end SingleLatticeCovering.Prekopa namespace SingleLatticeCovering.Prekopa open MeasureTheory Set Filter open scoped ENNReal Topology def ScalarLogConcave (f : ℝ → ℝ) : Prop := (∀ t, 0 ≤ f t) ∧ ∀ u v a b : ℝ, 0 < a → 0 < b → a+b=1 → f u^a*f v^b ≤ f (a*u+b*v) def linearCap {d : ℕ} (K : Set (RV d)) (L : RV d →L[ℝ] ℝ) (t : ℝ) : Set (RV d) := K ∩ {x | t ≤ L x} lemma linearCap_compact {d : ℕ} {K : Set (RV d)} (hK : IsCompact K) (L : RV d →L[ℝ] ℝ) (t : ℝ) : IsCompact (linearCap K L t) := hK.inter_right (isClosed_le continuous_const L.continuous) theorem scalarLogConcave_cap_volume {d : ℕ} {K : Set (RV d)} (hK : IsCompact K) (hconv : Convex ℝ K) (L : RV d →L[ℝ] ℝ) : ScalarLogConcave (fun t => (volume (linearCap K L t)).toReal) := by refine ⟨fun _ => ENNReal.toReal_nonneg,fun u v a b ha hb hab => ?_⟩ apply compact_volume (linearCap_compact hK L u) (linearCap_compact hK L v) (linearCap_compact hK L (a*u+b*v)) ha hb hab intro x hx y hy refine ⟨hconv hx.1 hy.1 ha.le hb.le hab,?_⟩ change a*u+b*v ≤ L (a•x+b•y) simp only [map_add,map_smul,smul_eq_mul] exact add_le_add (mul_le_mul_of_nonneg_left hx.2 ha.le) (mul_le_mul_of_nonneg_left hy.2 hb.le) lemma ScalarLogConcave.const_mul {f : ℝ → ℝ} (hf : ScalarLogConcave f) {c : ℝ} (hc : 0 ≤ c) : ScalarLogConcave (fun t => c*f t) := by refine ⟨fun t => mul_nonneg hc (hf.1 t),fun u v a b ha hb hab => ?_⟩ rcases hc.eq_or_lt with rfl | hc · simp only [zero_mul,Real.zero_rpow (ne_of_gt ha),Real.zero_rpow (ne_of_gt hb),le_refl] rw [Real.mul_rpow hc.le (hf.1 u),Real.mul_rpow hc.le (hf.1 v)] calc c^a*f u^a*(c^b*f v^b) = (c^a*c^b)*(f u^a*f v^b) := by ring _ = c*(f u^a*f v^b) := by rw [←Real.rpow_add hc,hab,Real.rpow_one] _ ≤ c*f (a*u+b*v) := mul_le_mul_of_nonneg_left (hf.2 u v a b ha hb hab) hc.le lemma ScalarLogConcave.log_interpolation {f : ℝ → ℝ} (hf : ScalarLogConcave f) {u v a b : ℝ} (hu : 0 < f u) (hv : 0 < f v) (ha : 0 < a) (hb : 0 < b) (hab : a+b=1) : a*Real.log (f u)+b*Real.log (f v) ≤ Real.log (f (a*u+b*v)) := by have hp : 0 < f u^a*f v^b := mul_pos (Real.rpow_pos_of_pos hu _) (Real.rpow_pos_of_pos hv _) have H := Real.log_le_log hp (hf.2 u v a b ha hb hab) rwa [Real.log_mul (ne_of_gt (Real.rpow_pos_of_pos hu _)) (ne_of_gt (Real.rpow_pos_of_pos hv _)), Real.log_rpow hu,Real.log_rpow hv] at H theorem ScalarLogConcave.exponential_tail {f : ℝ → ℝ} (hf : ScalarLogConcave f) (hle : ∀ t, f t ≤ 1) (hleft : (3:ℝ)/4 ≤ f (-2)) (hright : f 2 ≤ (1:ℝ)/4) {t : ℝ} (ht : 0 ≤ t) : f t ≤ Real.exp (1-t/16) := by have _ := ht by_cases ht2 : t ≤ 2 · exact (hle t).trans (Real.one_le_exp (by linarith)) have ht2 : 2 < t := lt_of_not_ge ht2 rcases (hf.1 t).eq_or_lt with hzero | hpos · rw [←hzero]; exact (Real.exp_pos _).le have hleftpos : 0 < f (-2) := by linarith let a := (t-2)/(t+2) let b := 4/(t+2) have htp : 0 < t+2 := by linarith have ha : 0 < a := div_pos (by linarith) htp have hb : 0 < b := div_pos (by norm_num) htp have hab : a+b=1 := by dsimp [a,b]; field_simp; ring have hinter : a*(-2)+b*t=2 := by dsimp [a,b]; field_simp; ring have H := hf.log_interpolation hleftpos hpos ha hb hab rw [hinter] at H have hl : -(1:ℝ)/2 ≤ Real.log (f (-2)) := by have A := Real.one_sub_inv_le_log_of_pos (show (0:ℝ) < 3/4 by norm_num) have B := Real.log_le_log (show (0:ℝ) < 3/4 by norm_num) hleft norm_num at A linarith have h2pos : 0 < f 2 := by have HH := hf.2 (-2) t a b ha hb hab rw [hinter] at HH exact (mul_pos (Real.rpow_pos_of_pos hleftpos _) (Real.rpow_pos_of_pos hpos _)).trans_le HH have hr : Real.log (f 2) ≤ -(3:ℝ)/4 := by have A := Real.log_le_log h2pos hright have B := Real.log_le_sub_one_of_pos (show (0:ℝ) < 1/4 by norm_num) linarith have H' : a*(-(1:ℝ)/2)+b*Real.log (f t) ≤ -(3:ℝ)/4 := by have A := mul_le_mul_of_nonneg_left hl ha.le linarith have H'' := mul_le_mul_of_nonneg_right H' htp.le have haeq : a*(t+2)=t-2 := div_mul_cancel₀ _ (ne_of_gt htp) have hbeq : b*(t+2)=4 := div_mul_cancel₀ _ (ne_of_gt htp) have hcalc : (a*(-(1:ℝ)/2)+b*Real.log (f t))*(t+2)= (t-2)*(-(1:ℝ)/2)+4*Real.log (f t) := by calc _ = (a*(t+2))*(-(1:ℝ)/2)+(b*(t+2))*Real.log (f t) := by ring _ = _ := by rw [haeq,hbeq] rw [hcalc] at H'' have hlog : Real.log (f t) ≤ 1-t/16 := by linarith simpa only [Real.exp_log hpos] using Real.exp_le_exp.mpr hlog end SingleLatticeCovering.Prekopa namespace SingleLatticeCovering.Prekopa open MeasureTheory Set Filter open scoped Topology lemma right_derivative_nonpos_of_max {F : ℝ → ℝ} {A : ℝ} (hF : HasDerivAt F A 0) (hmax : ∀ t ∈ Ioo (0:ℝ) 1, F t ≤ F 0) : A ≤ 0 := by apply le_of_tendsto hF.tendsto_slope_zero_right have hnear : ∀ᶠ t : ℝ in 𝓝[>] 0, t < 1 := nhdsWithin_le_nhds (Iio_mem_nhds (show (0:ℝ) < 1 by norm_num)) filter_upwards [self_mem_nhdsWithin,hnear] with t ht ht1 simp only [zero_add,smul_eq_mul] exact mul_nonpos_of_nonneg_of_nonpos (inv_nonneg.mpr (le_of_lt ht)) (sub_nonpos.mpr (hmax t ⟨ht,ht1⟩)) theorem ScalarLogConcave.log_bound_at_weighted_mode {φ : ℝ → ℝ} (hφ : ScalarLogConcave φ) {m : ℕ} {r₀ r : ℝ} (hr₀ : 0 < r₀) (hr : 0 < r) (hφ₀ : 0 < φ r₀) (hφr : 0 < φ r) (hmax : ∀ s : ℝ, 0 < s → s^m*φ s ≤ r₀^m*φ r₀) : Real.log (φ r)-Real.log (φ r₀) ≤ -(m:ℝ)*(r-r₀)/r₀ := by let F : ℝ → ℝ := fun t => (m:ℝ)*Real.log ((1-t)*r₀+t*r)+ (1-t)*Real.log (φ r₀)+t*Real.log (φ r) have hdlin : HasDerivAt (fun t : ℝ => (1-t)*r₀+t*r) (r-r₀) 0 := by convert! (((hasDerivAt_const (0:ℝ) (1:ℝ)).sub (hasDerivAt_id (0:ℝ))).mul_const r₀).add ((hasDerivAt_id (0:ℝ)).mul_const r) using 1 ; simp only [zero_sub,neg_one_mul,one_mul] ; ring have hdlog := hdlin.log (by simpa using ne_of_gt hr₀) have hd : HasDerivAt F ((m:ℝ)*(r-r₀)/r₀-Real.log (φ r₀)+Real.log (φ r)) 0 := by convert! ((hdlog.const_mul (m:ℝ)).add (((hasDerivAt_const (0:ℝ) (1:ℝ)).sub (hasDerivAt_id (0:ℝ))).mul_const (Real.log (φ r₀)))).add ((hasDerivAt_id (0:ℝ)).mul_const (Real.log (φ r))) using 1 ; simp only [sub_zero,one_mul,zero_mul,add_zero,zero_sub,neg_one_mul] ; ring have hFmax : ∀ t ∈ Ioo (0:ℝ) 1, F t ≤ F 0 := by intro t ht let z := (1-t)*r₀+t*r have hz : 0 < z := add_pos (mul_pos (by linarith [ht.2]) hr₀) (mul_pos ht.1 hr) have hinter := hφ.log_interpolation hφ₀ hφr (show 0 < 1-t by linarith [ht.2]) ht.1 (by ring) change (1-t)*Real.log (φ r₀)+t*Real.log (φ r) ≤ Real.log (φ z) at hinter have hφz : 0 < φ z := by have H := hφ.2 r₀ r (1-t) t (by linarith [ht.2]) ht.1 (by ring) exact (mul_pos (Real.rpow_pos_of_pos hφ₀ _) (Real.rpow_pos_of_pos hφr _)).trans_le H have H := Real.log_le_log (mul_pos (pow_pos hz _) hφz) (hmax z hz) rw [Real.log_mul (ne_of_gt (pow_pos hz _)) (ne_of_gt hφz), Real.log_mul (ne_of_gt (pow_pos hr₀ _)) (ne_of_gt hφ₀),Real.log_pow,Real.log_pow] at H dsimp [F] simp only [sub_zero,one_mul,zero_mul,add_zero] change (m:ℝ)*Real.log z+((1-t)*Real.log (φ r₀))+t*Real.log (φ r) ≤ _ linarith have H := right_derivative_nonpos_of_max hd hFmax rw [neg_mul,neg_div] linarith def shellRate (u : ℝ) : ℝ := u-1-Real.log u theorem ScalarLogConcave.weighted_mode_envelope {φ : ℝ → ℝ} (hφ : ScalarLogConcave φ) {m : ℕ} {r₀ r : ℝ} (hr₀ : 0 < r₀) (hr : 0 < r) (hφ₀ : 0 < φ r₀) (hmax : ∀ s : ℝ, 0 < s → s^m*φ s ≤ r₀^m*φ r₀) : r^m*φ r ≤ (r₀^m*φ r₀)*Real.exp (-(m:ℝ)*shellRate (r/r₀)) := by rcases (hφ.1 r).eq_or_lt with hz | hp · rw [←hz,mul_zero]; positivity have H := hφ.log_bound_at_weighted_mode hr₀ hr hφ₀ hp hmax have hlog : Real.log (r^m*φ r) ≤ Real.log (r₀^m*φ r₀)-(m:ℝ)*shellRate (r/r₀) := by rw [Real.log_mul (ne_of_gt (pow_pos hr _)) (ne_of_gt hp), Real.log_mul (ne_of_gt (pow_pos hr₀ _)) (ne_of_gt hφ₀),Real.log_pow,Real.log_pow] rw [shellRate,Real.log_div (ne_of_gt hr) (ne_of_gt hr₀)] have he : (r-r₀)/r₀=r/r₀-1 := by field_simp rw [neg_mul,neg_div,mul_div_assoc,he] at H nlinarith have HE := Real.exp_le_exp.mpr hlog rw [Real.exp_log (mul_pos (pow_pos hr _) hp),Real.exp_sub, Real.exp_log (mul_pos (pow_pos hr₀ _) hφ₀),div_eq_mul_inv,←Real.exp_neg] at HE simpa only [neg_mul] using HE end SingleLatticeCovering.Prekopa namespace SingleLatticeCovering.Prekopa open MeasureTheory Set Filter open scoped Topology lemma shellRate_ge_sqrt_sq {u : ℝ} (hu : 0 < u) : (Real.sqrt u-1)^2 ≤ shellRate u := by have H := Real.log_le_sub_one_of_pos (Real.sqrt_pos.mpr hu) rw [Real.log_sqrt hu.le] at H dsimp [shellRate] nlinarith [Real.sq_sqrt hu.le] lemma shellRate_ge_sq_div_nine {u : ℝ} (hu : 0 < u) (hu2 : u ≤ 2) : (u-1)^2/9 ≤ shellRate u := by have hs := Real.sqrt_nonneg u have hs2 : Real.sqrt u ≤ 2 := by nlinarith [Real.sq_sqrt hu.le] have he : (u-1)^2=(Real.sqrt u-1)^2*(Real.sqrt u+1)^2 := by calc _ = ((Real.sqrt u)^2-1)^2 := by rw [Real.sq_sqrt hu.le] _ = _ := by ring have hsq : (Real.sqrt u+1)^2 ≤ 9 := by nlinarith have hm := mul_le_mul_of_nonneg_left hsq (sq_nonneg (Real.sqrt u-1)) rw [←he] at hm nlinarith [shellRate_ge_sqrt_sq hu] lemma shellRate_support {u a : ℝ} (hu : 0 < u) (ha : 0 < a) : shellRate a+(1-a⁻¹)*(u-a) ≤ shellRate u := by have H := Real.log_le_sub_one_of_pos (div_pos hu ha) rw [Real.log_div (ne_of_gt hu) (ne_of_gt ha)] at H have he : (1-a⁻¹)*(u-a)=u-a-u/a+1 := by field_simp; ring rw [he] dsimp [shellRate] linarith lemma shellRate_upper_tail {ε u : ℝ} (hε : 0 < ε) (hε1 : ε ≤ 1) (hu : 1+ε ≤ u) : ε^2/9+(ε/2)*(u-(1+ε)) ≤ shellRate u := by have ha : 0 < 1+ε := by linarith have hs := shellRate_support (show 0 < u by linarith) ha have hb := shellRate_ge_sq_div_nine ha (show 1+ε ≤ 2 by linarith) have hc : ε/2 ≤ 1-(1+ε)⁻¹ := by have he : (1-(1+ε)⁻¹)*(1+ε)=ε := by rw [sub_mul,one_mul,inv_mul_cancel₀ (ne_of_gt ha)] ring apply (mul_le_mul_iff_left₀ ha).mp nlinarith have hm := mul_le_mul_of_nonneg_right hc (sub_nonneg.mpr hu) nlinarith lemma shellRate_lower_tail {ε u : ℝ} (hε : 0 < ε) (hε1 : ε < 1) (hu : 0 < u) (hu1 : u ≤ 1-ε) : ε^2/9+(ε/2)*((1-ε)-u) ≤ shellRate u := by have ha : 0 < 1-ε := by linarith have hs := shellRate_support hu ha have hb := shellRate_ge_sq_div_nine ha (show 1-ε ≤ 2 by linarith) have hc : 1-(1-ε)⁻¹ ≤ -ε/2 := by have he : (1-(1-ε)⁻¹)*(1-ε) = -ε := by rw [sub_mul,one_mul,inv_mul_cancel₀ (ne_of_gt ha)] ring apply (mul_le_mul_iff_left₀ ha).mp nlinarith have hm := mul_le_mul_of_nonpos_right hc (sub_nonpos.mpr hu1) nlinarith end SingleLatticeCovering.Prekopa namespace SingleLatticeCovering.Prekopa open MeasureTheory Set Filter open scoped Topology lemma integral_exp_shift_Ioi {a : ℝ} (ha : 0 < a) (c : ℝ) : (∫ r : ℝ in Ioi c, Real.exp (-a*(r-c)))=a⁻¹ := by have he (r : ℝ) : Real.exp (-a*(r-c))=Real.exp (a*c)*Real.exp (-a*r) := by rw [←Real.exp_add]; congr 1; ring simp_rw [he] rw [integral_const_mul,integral_exp_mul_Ioi (by linarith)] rw [neg_div_neg_eq,div_eq_mul_inv,←mul_assoc,←Real.exp_add] simp lemma integral_exp_shift_Iic {a : ℝ} (ha : 0 < a) (c : ℝ) : (∫ r : ℝ in Iic c, Real.exp (a*(r-c)))=a⁻¹ := by have he (r : ℝ) : Real.exp (a*(r-c))=Real.exp (-a*c)*Real.exp (a*r) := by rw [←Real.exp_add]; congr 1; ring simp_rw [he] rw [integral_const_mul,integral_exp_mul_Iic ha] rw [div_eq_mul_inv,←mul_assoc,←Real.exp_add] simp lemma integrableOn_exp_shift_Ioi {a : ℝ} (ha : 0 < a) (c : ℝ) : IntegrableOn (fun r : ℝ => Real.exp (-a*(r-c))) (Ioi c) volume := by have he (r : ℝ) : Real.exp (-a*(r-c))=Real.exp (a*c)*Real.exp (-a*r) := by rw [←Real.exp_add]; congr 1; ring simp_rw [he] exact (integrableOn_exp_mul_Ioi (by linarith) c).const_mul _ lemma integrableOn_exp_shift_Iic {a : ℝ} (ha : 0 < a) (c : ℝ) : IntegrableOn (fun r : ℝ => Real.exp (a*(r-c))) (Iic c) volume := by have he (r : ℝ) : Real.exp (a*(r-c))=Real.exp (-a*c)*Real.exp (a*r) := by rw [←Real.exp_add]; congr 1; ring simp_rw [he] exact (integrableOn_exp_mul_Iic ha c).const_mul _ lemma ScalarLogConcave.upper_shell_envelope {φ : ℝ → ℝ} (hφ : ScalarLogConcave φ) {m : ℕ} {r₀ ε r : ℝ} (hr₀ : 0 < r₀) (hε : 0 < ε) (hε1 : ε ≤ 1) (hφ₀ : 0 < φ r₀) (hmax : ∀ s : ℝ, 0 < s → s^m*φ s ≤ r₀^m*φ r₀) (hr : (1+ε)*r₀ ≤ r) : r^m*φ r ≤ (r₀^m*φ r₀*Real.exp (-(m:ℝ)*ε^2/9))* Real.exp (-((m:ℝ)*ε/(2*r₀))*(r-(1+ε)*r₀)) := by have hrp : 0 < r := (mul_pos (by linarith) hr₀).trans_le hr have hu : 1+ε ≤ r/r₀ := (le_div_iff₀ hr₀).mpr hr have H := mul_le_mul_of_nonpos_left (shellRate_upper_tail hε hε1 hu) (neg_nonpos.mpr (Nat.cast_nonneg m)) have E := hφ.weighted_mode_envelope hr₀ hrp hφ₀ hmax apply E.trans rw [mul_assoc (r₀^m*φ r₀),←Real.exp_add] apply mul_le_mul_of_nonneg_left _ (mul_nonneg (pow_nonneg hr₀.le _) hφ₀.le) apply Real.exp_le_exp.mpr exact H.trans_eq (by field_simp; ring) lemma ScalarLogConcave.lower_shell_envelope {φ : ℝ → ℝ} (hφ : ScalarLogConcave φ) {m : ℕ} {r₀ ε r : ℝ} (hr₀ : 0 < r₀) (hε : 0 < ε) (hε1 : ε < 1) (hφ₀ : 0 < φ r₀) (hmax : ∀ s : ℝ, 0 < s → s^m*φ s ≤ r₀^m*φ r₀) (hr : 0 < r) (hr1 : r ≤ (1-ε)*r₀) : r^m*φ r ≤ (r₀^m*φ r₀*Real.exp (-(m:ℝ)*ε^2/9))* Real.exp (((m:ℝ)*ε/(2*r₀))*(r-(1-ε)*r₀)) := by have hu : r/r₀ ≤ 1-ε := (div_le_iff₀ hr₀).mpr hr1 have H := mul_le_mul_of_nonpos_left (shellRate_lower_tail hε hε1 (div_pos hr hr₀) hu) (neg_nonpos.mpr (Nat.cast_nonneg m)) have E := hφ.weighted_mode_envelope hr₀ hr hφ₀ hmax apply E.trans rw [mul_assoc (r₀^m*φ r₀),←Real.exp_add] apply mul_le_mul_of_nonneg_left _ (mul_nonneg (pow_nonneg hr₀.le _) hφ₀.le) apply Real.exp_le_exp.mpr exact H.trans_eq (by field_simp; ring) lemma ScalarLogConcave.upper_shell_integral {φ : ℝ → ℝ} (hφ : ScalarLogConcave φ) {m : ℕ} (hm : 0 < m) {r₀ ε : ℝ} (hr₀ : 0 < r₀) (hε : 0 < ε) (hε1 : ε ≤ 1) (hφ₀ : 0 < φ r₀) (hmax : ∀ s : ℝ, 0 < s → s^m*φ s ≤ r₀^m*φ r₀) (hi : IntegrableOn (fun r => r^m*φ r) (Ioi 0) volume) : (∫ r in Ioi ((1+ε)*r₀), r^m*φ r) ≤ (r₀^m*φ r₀*Real.exp (-(m:ℝ)*ε^2/9))*((m:ℝ)*ε/(2*r₀))⁻¹ := by have ha : 0 < (m:ℝ)*ε/(2*r₀) := by positivity calc _ ≤ ∫ r in Ioi ((1+ε)*r₀), (r₀^m*φ r₀*Real.exp (-(m:ℝ)*ε^2/9))*Real.exp (-((m:ℝ)*ε/(2*r₀))*(r-(1+ε)*r₀)) := by have hs : Ioi ((1+ε)*r₀) ⊆ Ioi 0 := by intro r hr exact ((mul_pos (by linarith) hr₀).trans hr) apply integral_mono_ae (hi.mono_set hs) ((integrableOn_exp_shift_Ioi ha _).const_mul _) filter_upwards [ae_restrict_mem measurableSet_Ioi] with r hr exact hφ.upper_shell_envelope hr₀ hε hε1 hφ₀ hmax (le_of_lt hr) _ = _ := by rw [integral_const_mul,integral_exp_shift_Ioi ha] lemma ScalarLogConcave.lower_shell_integral {φ : ℝ → ℝ} (hφ : ScalarLogConcave φ) {m : ℕ} (hm : 0 < m) {r₀ ε : ℝ} (hr₀ : 0 < r₀) (hε : 0 < ε) (hε1 : ε < 1) (hφ₀ : 0 < φ r₀) (hmax : ∀ s : ℝ, 0 < s → s^m*φ s ≤ r₀^m*φ r₀) (hi : IntegrableOn (fun r => r^m*φ r) (Ioi 0) volume) : (∫ r in Ioc 0 ((1-ε)*r₀), r^m*φ r) ≤ (r₀^m*φ r₀*Real.exp (-(m:ℝ)*ε^2/9))*((m:ℝ)*ε/(2*r₀))⁻¹ := by have ha : 0 < (m:ℝ)*ε/(2*r₀) := by positivity let F : ℝ → ℝ := fun r => (r₀^m*φ r₀*Real.exp (-(m:ℝ)*ε^2/9))* Real.exp (((m:ℝ)*ε/(2*r₀))*(r-(1-ε)*r₀)) have hF : IntegrableOn F (Iic ((1-ε)*r₀)) volume := (integrableOn_exp_shift_Iic ha _).const_mul _ calc _ ≤ ∫ r in Ioc 0 ((1-ε)*r₀), F r := by apply integral_mono_ae (hi.mono_set Ioc_subset_Ioi_self) (hF.mono_set Ioc_subset_Iic_self) filter_upwards [ae_restrict_mem measurableSet_Ioc] with r hr exact hφ.lower_shell_envelope hr₀ hε hε1 hφ₀ hmax hr.1 hr.2 _ ≤ ∫ r in Iic ((1-ε)*r₀), F r := by apply setIntegral_mono_set hF · exact Filter.Eventually.of_forall (fun r => by dsimp [F]; positivity) · exact Filter.Eventually.of_forall Ioc_subset_Iic_self _ = _ := by dsimp [F]; rw [integral_const_mul,integral_exp_shift_Iic ha] lemma radial_integral_lower {φ : ℝ → ℝ} {m : ℕ} {r₀ : ℝ} (hr₀ : 0 < r₀) (hφ : ∀ r, 0 ≤ φ r) (hmono : ∀ r ∈ Ioc 0 r₀, φ r₀ ≤ φ r) (hi : IntegrableOn (fun r => r^m*φ r) (Ioi 0) volume) : r₀^m*φ r₀*r₀/(m+1:ℝ) ≤ ∫ r in Ioi 0, r^m*φ r := by have hpoly : IntegrableOn (fun r : ℝ => r^m*φ r₀) (Ioc 0 r₀) volume := (Continuous.integrableOn_Icc (by fun_prop)).mono_set Ioc_subset_Icc_self have hcalc : (∫ r in Ioc 0 r₀, r^m*φ r₀)=r₀^m*φ r₀*r₀/(m+1:ℝ) := by rw [←intervalIntegral.integral_of_le hr₀.le,intervalIntegral.integral_mul_const,integral_pow] simp only [zero_pow (Nat.succ_ne_zero m),sub_zero,pow_succ] ring rw [←hcalc] calc _ ≤ ∫ r in Ioc 0 r₀, r^m*φ r := by apply integral_mono_ae hpoly (hi.mono_set Ioc_subset_Ioi_self) filter_upwards [ae_restrict_mem measurableSet_Ioc] with r hr exact mul_le_mul_of_nonneg_left (hmono r hr) (pow_nonneg hr.1.le _) _ ≤ _ := by apply setIntegral_mono_set hi · filter_upwards [ae_restrict_mem measurableSet_Ioi] with r hr exact mul_nonneg (pow_nonneg hr.le _) (hφ r) · exact Filter.Eventually.of_forall Ioc_subset_Ioi_self theorem ScalarLogConcave.shell_integral_bound {φ : ℝ → ℝ} (hφ : ScalarLogConcave φ) {m : ℕ} (hm : 0 < m) {r₀ ε : ℝ} (hr₀ : 0 < r₀) (hε : 0 < ε) (hε1 : ε < 1) (hφ₀ : 0 < φ r₀) (hmax : ∀ s : ℝ, 0 < s → s^m*φ s ≤ r₀^m*φ r₀) (hmono : ∀ r ∈ Ioc 0 r₀, φ r₀ ≤ φ r) (hi : IntegrableOn (fun r => r^m*φ r) (Ioi 0) volume) : (∫ r in Ioc 0 ((1-ε)*r₀), r^m*φ r)+ (∫ r in Ioi ((1+ε)*r₀), r^m*φ r) ≤ (4*((m:ℝ)+1)/((m:ℝ)*ε)*Real.exp (-(m:ℝ)*ε^2/9))*(∫ r in Ioi 0, r^m*φ r) := by have HL := hφ.lower_shell_integral hm hr₀ hε hε1 hφ₀ hmax hi have HU := hφ.upper_shell_integral hm hr₀ hε hε1.le hφ₀ hmax hi have HN := radial_integral_lower hr₀ hφ.1 hmono hi have hm' : 0 < (m:ℝ) := Nat.cast_pos.mpr hm have HN' : r₀^m*φ r₀*r₀ ≤ ((m:ℝ)+1)*(∫ r in Ioi 0, r^m*φ r) := by have H := (div_le_iff₀ (show 0 < (m:ℝ)+1 by positivity)).mp HN linarith have he : 2*((r₀^m*φ r₀*Real.exp (-(m:ℝ)*ε^2/9))*((m:ℝ)*ε/(2*r₀))⁻¹)= (4/((m:ℝ)*ε)*Real.exp (-(m:ℝ)*ε^2/9))*(r₀^m*φ r₀*r₀) := by field_simp; ring have HE := mul_le_mul_of_nonneg_left HN' (show 0 ≤ 4/((m:ℝ)*ε)*Real.exp (-(m:ℝ)*ε^2/9) by positivity) calc _ ≤ 2*((r₀^m*φ r₀*Real.exp (-(m:ℝ)*ε^2/9))*((m:ℝ)*ε/(2*r₀))⁻¹) := by linarith _ = _ := he _ ≤ _ := HE _ = _ := by ring lemma exists_weighted_mode {φ : ℝ → ℝ} {m : ℕ} (hm : 0 < m) (hc : Continuous φ) (hp : 0 < φ 1) (ht : Tendsto (fun r : ℝ => r^m*φ r) atTop (𝓝 0)) : ∃ r₀ : ℝ, 0 < r₀ ∧ 0 < φ r₀ ∧ ∀ r : ℝ, 0 < r → r^m*φ r ≤ r₀^m*φ r₀ := by have hev : ∀ᶠ r : ℝ in atTop, r^m*φ r < φ 1 := ht (Iio_mem_nhds hp) obtain ⟨R,hR⟩ := eventually_atTop.mp hev let B := max R 1 have hB : (1:ℝ) ≤ B := le_max_right _ _ have hcont : Continuous (fun r : ℝ => r^m*φ r) := by fun_prop obtain ⟨r₀,hr₀,hmax⟩ := isCompact_Icc.exists_isMaxOn (nonempty_Icc.mpr (by linarith : (0:ℝ) ≤ B)) hcont.continuousOn have hbase : φ 1 ≤ r₀^m*φ r₀ := by simpa only [Set.mem_ofPred_eq,one_pow,one_mul] using hmax (show (1:ℝ) ∈ Icc 0 B from ⟨by norm_num,hB⟩) have hpos : 0 < r₀ := by rcases hr₀.1.eq_or_lt with hz | hz · subst r₀ simp only [zero_pow (Nat.ne_of_gt hm),zero_mul] at hbase linarith · exact hz have hφpos : 0 < φ r₀ := (mul_pos_iff_of_pos_left (pow_pos hpos m)).mp (hp.trans_le hbase) refine ⟨r₀,hpos,hφpos,fun r hr => ?_⟩ by_cases hrB : r ≤ B · exact hmax ⟨hr.le,hrB⟩ · exact (hR r ((le_max_left R 1).trans (le_of_not_ge hrB))).le.trans hbase end SingleLatticeCovering.Prekopa end end end end end end OAI