| import Mathlib |
| import OAI.Geometry.DoublingHilbert.CompactModel |
| import OAI.Geometry.DoublingHilbert.FiniteWitness |
| import OAI.Geometry.DoublingHilbert.Auerbach |
| import OAI.Geometry.DoublingHilbert.ClusterPlacement |
| import OAI.Geometry.DoublingHilbert.ClusterCover |
| import OAI.Geometry.DoublingHilbert.FiniteTargets |
| import OAI.Geometry.DoublingHilbert.Main |
|
|
| namespace OAI |
|
|
| open Set Metric |
| open scoped BigOperators |
| open Filter |
| open scoped Topology |
| noncomputable section |
| universe u v |
| namespace CompactBanach |
|
|
| theorem main : ∃ Λ : ℕ, ∀ (B : Type u) [NormedAddCommGroup B] [NormedSpace ℝ B] |
| [CompleteSpace B], ¬ FiniteDimensional ℝ B → |
| ∃ K : Set B, IsCompact K ∧ DoublingAtMost K Λ ∧ |
| ∀ (E : Type v) [NormedAddCommGroup E] [NormedSpace ℝ E] |
| [FiniteDimensional ℝ E], ¬ AdmitsBiLipschitzEmbedding K E := by |
| classical |
| refine ⟨1 + 5 * (76800 ^ 8) ^ 2, ?_⟩ |
| intro B _ _ _ hB |
| have hS : AmbientDoubling DoublingHilbert.constructedSet 76800 := |
| doubling_iff_ambient.mp (fun x r hr => DoublingHilbert.constructedSet_cover x hr) |
| |
| have hw (j : ℕ) := finite_witness_card (Nat.unpair j).1 |
| (A := ((Nat.unpair j).2 : ℝ) + 1) (le_add_of_nonneg_left (Nat.cast_nonneg _)) |
| choose X hXcard hX using hw |
| obtain ⟨v, hv, hvcoord⟩ := exists_auerbach_vectors (ι := Unit) hB |
| have hvunit : ‖v ()‖ = 1 := by |
| apply le_antisymm (hv ()) |
| simpa using hvcoord (fun _ => 1) () |
| have hp (j : ℕ) := cluster_placement hB hS (X j) (v ()) (clusterScale_pos j) |
| choose f hloc hdoubling a ha hd using hp |
| let C : ℕ → Finset B := fun j => Finset.univ.image (f j) |
| have hCset (j : ℕ) : (C j : Set B) = Set.range (f j) := by |
| ext z |
| simp [C] |
| have hCloc : ∀ j z, z ∈ C j → |
| dist z (clusterScale j • v ()) ≤ clusterScale j / 100 := by |
| intro j z hz |
| obtain ⟨x, _, rfl⟩ := Finset.mem_image.mp hz |
| exact hloc j x |
| have hCdouble : ∀ j, DoublingAtMost (C j : Set B) (76800 ^ 8) := by |
| intro j |
| rw [hCset] |
| exact hdoubling j |
| let K : Set B := {0} ∪ ⋃ j : ℕ, (C j : Set B) |
| obtain ⟨hcompact, hdouble⟩ := compact_doubling_clusters C hvunit hCloc hCdouble |
| refine ⟨K, hcompact, hdouble, ?_⟩ |
| intro E _ _ _ hF |
| obtain ⟨F, b, hb, D, hD, hF⟩ := hF |
| obtain ⟨g, c, hc, A, hA, hg⟩ := euclidean_embedding_of_finite_dimensional F hb hD hF |
| obtain ⟨n, hn⟩ := exists_nat_gt (36 * A) |
| let j : ℕ := Nat.pair (Module.finrank ℝ E) n |
| have hjdim : (Nat.unpair j).1 = Module.finrank ℝ E := by simp [j] |
| have hjn : (Nat.unpair j).2 = n := by simp [j] |
| let intoK : X j → K := fun x => |
| ⟨f j x, Or.inr (Set.mem_iUnion.mpr ⟨j, |
| Finset.mem_image.mpr ⟨x, Finset.mem_univ _, rfl⟩⟩)⟩ |
| have hbad := hX j |
| rw [hjdim, hjn] at hbad |
| apply hbad |
| refine ⟨g ∘ intoK, c * a j, mul_pos hc (ha j), ?_⟩ |
| intro x y |
| have hgg := hg (intoK x) (intoK y) |
| change c * dist (f j x) (f j y) ≤ dist (g (intoK x)) (g (intoK y)) ∧ |
| dist (g (intoK x)) (g (intoK y)) ≤ A * c * dist (f j x) (f j y) at hgg |
| dsimp only [Function.comp_apply] |
| constructor |
| · calc |
| _ = c * (a j * dist x y) := by ring |
| _ ≤ c * dist (f j x) (f j y) := mul_le_mul_of_nonneg_left (hd j x y).1 hc.le |
| _ ≤ _ := hgg.1 |
| · have hApos : 0 ≤ A := le_trans zero_le_one hA |
| have hn' : 36 * A ≤ (n : ℝ) + 1 := by linarith |
| calc |
| _ ≤ A * c * dist (f j x) (f j y) := hgg.2 |
| _ ≤ A * c * (36 * a j * dist x y) := |
| mul_le_mul_of_nonneg_left (hd j x y).2 (mul_nonneg hApos hc.le) |
| _ = (36 * A) * (c * a j) * dist x y := by ring |
| _ ≤ ((n : ℝ) + 1) * (c * a j) * dist x y := |
| mul_le_mul_of_nonneg_right |
| (mul_le_mul_of_nonneg_right hn' (mul_nonneg hc.le (ha j).le)) dist_nonneg |
|
|
| end CompactBanach |
| end |
|
|
| end OAI |
|
|
