| import Mathlib |
|
|
| namespace OAI |
|
|
| noncomputable section |
| open Set Filter Manifold Bundle MeasureTheory |
| open scoped Topology ContDiff ENNReal |
| open Set Filter Manifold Bundle |
| open scoped Topology ContDiff |
| open Set Filter Metric |
| open scoped Topology InnerProductSpace |
| open Set Filter Function Metric |
| open scoped Topology |
| open Set Filter Function Metric |
| open scoped Topology |
| open Set Filter Manifold |
| open scoped Topology ContDiff |
| open Set Filter MeasureTheory Metric |
| open scoped Topology ENNReal NNReal |
| open Set Filter Manifold Bundle MeasureTheory |
| open scoped Topology ContDiff ENNReal |
| open Set Filter Manifold Bundle |
| open scoped Topology ContDiff |
| open Set Filter Metric |
| open scoped Topology InnerProductSpace |
| open Set Filter Function Metric |
| open scoped Topology |
| open Set Filter Function Metric |
| open scoped Topology |
| open Set Filter |
| namespace YauCounterexamples |
| lemma compact_strict_profile_ratio {X : Type*} [TopologicalSpace X] |
| {K : Set X} (hK : IsCompact K) {r φ : X → ℝ} |
| (hr : ContinuousOn r K) (hφ : ContinuousOn φ K) |
| (hr0 : ∀ y ∈ K, 0 ≤ r y) |
| (hgap : ∀ y ∈ K, r y ≠ 0 → Real.log (r y) < φ y) : |
| ∃ ρ : ℝ, 0 < ρ ∧ ρ < 1 ∧ ∀ y ∈ K, r y ≤ ρ*Real.exp (φ y) := by |
| classical |
| by_cases hne : K.Nonempty |
| · let f := fun y => r y / Real.exp (φ y) |
| have hf : ContinuousOn f K := hr.div hφ.rexp (fun y _ => (Real.exp_pos _).ne') |
| obtain ⟨z,hz,hm⟩ := hK.exists_isMaxOn hne hf |
| have hlt : f z < 1 := by |
| dsimp [f] |
| apply (div_lt_one (Real.exp_pos _)).mpr |
| by_cases hz0 : r z=0 |
| · rw [hz0]; exact Real.exp_pos _ |
| · have hrpos : 0 < r z := lt_of_le_of_ne (hr0 z hz) (Ne.symm hz0) |
| rw [←Real.exp_log hrpos] |
| exact Real.exp_lt_exp.mpr (hgap z hz hz0) |
| have hnonneg : 0 ≤ f z := div_nonneg (hr0 z hz) (Real.exp_pos _).le |
| refine ⟨(f z+1)/2,by linarith,by linarith,?_⟩ |
| intro y hy |
| apply (div_le_iff₀ (Real.exp_pos (φ y))).mp |
| have hh : f y ≤ f z := hm hy |
| change f y ≤ _ |
| linarith |
| · refine ⟨1/2,by norm_num,by norm_num,?_⟩ |
| intro y hy |
| exact (hne ⟨y,hy⟩).elim |
| |
| end YauCounterexamples |
| |
| end |
| |
| end OAI |
| |
