| import Mathlib | |
| namespace OAI | |
| noncomputable section | |
| open Set Filter | |
| open scoped Topology ContDiff | |
| namespace WeakMTWTransport | |
| variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] | |
| lemma first_order_remainder_le_of_hessian_bound {f : E → ℝ} {S : Set E} | |
| (hS : Convex ℝ S) {B : ℝ} (hB : 0≤B) | |
| (hf : ∀ z∈S, ContDiffAt ℝ 2 f z) | |
| (hbound : ∀ z∈S, ‖fderiv ℝ (fderiv ℝ f) z‖≤B) | |
| {x y : E} (hx : x∈S) (hy : y∈S) : | |
| |f y-f x-fderiv ℝ f x (y-x)|≤B*‖y-x‖^2 := by | |
| have hD : ∀ z∈S, HasFDerivAt (fderiv ℝ f) (fderiv ℝ (fderiv ℝ f) z) z := by | |
| intro z hz | |
| exact ((hf z hz).fderiv_right (m := 1) (by norm_num)).differentiableAt (by norm_num) |>.hasFDerivAt | |
| have hLip : ∀ z∈S, ‖fderiv ℝ f z-fderiv ℝ f x‖≤B*‖z-x‖ := by | |
| intro z hz | |
| exact hS.norm_image_sub_le_of_norm_hasFDerivWithin_le | |
| (fun w hw => (hD w hw).hasFDerivWithinAt) hbound hx hz | |
| let R := fun z => f z-f x-fderiv ℝ f x (z-x) | |
| have hR : ∀ z∈S, HasFDerivAt R (fderiv ℝ f z-fderiv ℝ f x) z := by | |
| intro z hz | |
| exact ((hf z hz).differentiableAt (by norm_num)).hasFDerivAt.sub_const (f x) |>.sub | |
| ((fderiv ℝ f x).hasFDerivAt.comp z ((hasFDerivAt_id z).sub_const x)) | |
| have hseg : segment ℝ x y ⊆ S := hS.segment_subset hx hy | |
| have hn : ∀ z∈segment ℝ x y, ‖z-x‖≤‖y-x‖ := by | |
| intro z hz | |
| simpa only [Metric.mem_closedBall,dist_eq_norm] using | |
| (show segment ℝ x y ⊆ Metric.closedBall x ‖y-x‖ from | |
| (convex_closedBall x ‖y-x‖).segment_subset | |
| (Metric.mem_closedBall_self (norm_nonneg _)) (by simp only [Metric.mem_closedBall,dist_eq_norm,le_refl])) hz | |
| have H := (convex_segment x y).norm_image_sub_le_of_norm_hasFDerivWithin_le | |
| (fun z hz => (hR z (hseg hz)).hasFDerivWithinAt) | |
| (show ∀ z∈segment ℝ x y, ‖fderiv ℝ f z-fderiv ℝ f x‖≤B*‖y-x‖ from by | |
| intro z hz | |
| exact (hLip z (hseg hz)).trans (mul_le_mul_of_nonneg_left (hn z hz) hB)) | |
| (left_mem_segment ℝ x y) (right_mem_segment ℝ x y) | |
| simpa only [R,sub_self,map_zero,sub_zero,Real.norm_eq_abs,pow_two,mul_assoc] using H | |
| lemma convexOn_of_affine_lower_supports {f : E → ℝ} {S : Set E} | |
| (hS : Convex ℝ S) | |
| (hs : ∀ x∈S, ∃ L : E →L[ℝ] ℝ, ∀ y∈S, f x+L (y-x)≤f y) : | |
| ConvexOn ℝ S f := by | |
| refine ⟨hS,?_⟩ | |
| intro x hx y hy a b ha hb hab | |
| let z := a • x+b • y | |
| obtain ⟨L,hL⟩ := hs z (hS hx hy ha hb hab) | |
| have hx' := mul_le_mul_of_nonneg_left (hL x hx) ha | |
| have hy' := mul_le_mul_of_nonneg_left (hL y hy) hb | |
| have hzero : a*L (x-z)+b*L (y-z)=0 := by | |
| rw [←smul_eq_mul a,←smul_eq_mul b,←map_smul,←map_smul,←map_add] | |
| have he : a • (x-z)+b • (y-z)=0 := by | |
| rw [smul_sub,smul_sub,←add_sub_add_comm,←add_smul,hab,one_smul] | |
| exact sub_self z | |
| rw [he,map_zero] | |
| change f z≤a*f x+b*f y | |
| have hc : a*f z+b*f z=f z := by rw [←add_mul,hab,one_mul] | |
| nlinarith [hc] | |
| end WeakMTWTransport | |
| end | |
| end OAI | |