| import OAI.Analysis.Daugavet.EquivalentNorm |
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| namespace OAI |
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| noncomputable section |
| open MeasureTheory Filter Set |
| open scoped ENNReal NNReal Topology |
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| universe uX u_1 |
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| namespace ExactModuli |
| variable {X : Type uX} [NormedAddCommGroup X] [NormedSpace ℝ X] |
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| /-- The renorming obstruction is a consequence of the weak-neighborhood geometry, not merely |
| of the vanishing inherited-norm one-sided modulus. -/ |
| theorem no_equivalent_auc (hinf : ¬ FiniteDimensional ℝ X) (hsp : UnitSliceProperty X) |
| (N : Seminorm ℝ X) (hN : EquivalentNorm N) : ¬ AUC N := by |
| obtain ⟨R, hRpos, hRball, hRapprox⟩ := equivalentNorm_supremum hinf hN |
| obtain ⟨a, b, ha, hb, hab⟩ := hN |
| have hN' : EquivalentNorm N := ⟨a, b, ha, hb, hab⟩ |
| let t := a / (4 * b) |
| have ht : 0 < t := by dsimp [t]; positivity |
| have hmod : oneSidedModulus N t ≤ 0 := by |
| apply le_of_forall_pos_le_add |
| intro ε hε |
| obtain ⟨x, hx, hopt⟩ := hRapprox ε hε |
| have hNx : 0 < N x := by have := (hab x).1; rw [hx, mul_one] at this; linarith |
| have hxN : N ((N x)⁻¹ • x) = 1 := seminorm_normalize N hNx |
| have hD := normD_bound_at_original_unit hsp N ha hb hab hRball hx |
| have hopt' : R < N x * (1 + ε) := (div_lt_iff₀ (by linarith)).mp hopt |
| have hquot : R / N x - 1 ≤ ε := by |
| have : R / N x < 1 + ε := (div_lt_iff₀ hNx).mpr (by nlinarith) |
| linarith |
| have hinf := csInf_le (normD_centers_bddBelow hinf hN' ht.le) |
| ⟨(N x)⁻¹ • x, hxN, rfl⟩ |
| change oneSidedModulus N t ≤ D N ((N x)⁻¹ • x) t at hinf |
| change D N ((N x)⁻¹ • x) t ≤ R / N x - 1 at hD |
| simpa only [zero_add] using hinf.trans (hD.trans hquot) |
| intro hauc |
| exact (hauc t ht).not_ge hmod |
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| theorem exact_main {Ω : Type u_1} [MeasurableSpace Ω] (μ : Measure Ω) |
| [IsProbabilityMeasure μ] (E : Submodule ℝ (Lp ℝ 1 μ)) |
| (_hclosed : IsClosed (E : Set (Lp ℝ 1 μ))) (hdim : ¬ FiniteDimensional ℝ E) |
| (hprecompact : MeasurePrecompactBall μ E) (hdaugavet : Daugavet E) : |
| (∀ (x : E), ‖x‖ = 1 → ∀ t : ℝ, 0 < t → |
| H (normSeminorm ℝ E) x t = max (t / 2) (t - 1) ∧ |
| D (normSeminorm ℝ E) x t = max 0 (t - 2)) ∧ |
| (∀ t : ℝ, 0 < t → |
| averagedModulus (normSeminorm ℝ E) t = max (t / 2) (t - 1) ∧ |
| oneSidedModulus (normSeminorm ℝ E) t = max 0 (t - 2)) ∧ |
| (∀ N : Seminorm ℝ E, EquivalentNorm N → ¬ AUC N) := by |
| exact ⟨fun x hx t ht => exact_pointwise E hdim hprecompact hdaugavet hx ht, |
| fun t ht => exact_center_infima E hdim hprecompact hdaugavet ht, |
| no_equivalent_auc hdim (daugavet_unitSlice hdaugavet)⟩ |
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| end ExactModuli |
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| end |
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| end OAI |
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