| import Mathlib |
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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 |
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| variable {X : Type uX} [NormedAddCommGroup X] [NormedSpace ℝ X] |
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| / |
| def Cofinite (F : Submodule ℝ X) : Prop := |
| IsClosed (F : Set X) ∧ FiniteDimensional ℝ (X ⧸ F) |
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| / |
| def averagedIncrement (N : Seminorm ℝ X) (x : X) (t : ℝ) (y : X) : ℝ := |
| (N (x + t • y) + N (x - t • y)) / 2 - 1 |
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| def H (N : Seminorm ℝ X) (x : X) (t : ℝ) : ℝ := |
| sSup {r | ∃ F : Submodule ℝ X, Cofinite F ∧ |
| r = sInf {a | ∃ y ∈ F, 1 ≤ N y ∧ a = averagedIncrement N x t y}} |
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| def D (N : Seminorm ℝ X) (x : X) (t : ℝ) : ℝ := |
| sSup {r | ∃ F : Submodule ℝ X, Cofinite F ∧ |
| r = sInf {a | ∃ y ∈ F, N y = 1 ∧ a = N (x + t • y) - 1}} |
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| def averagedModulus (N : Seminorm ℝ X) (t : ℝ) : ℝ := |
| sInf {r | ∃ x, N x = 1 ∧ r = H N x t} |
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| def oneSidedModulus (N : Seminorm ℝ X) (t : ℝ) : ℝ := |
| sInf {r | ∃ x, N x = 1 ∧ r = D N x t} |
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| / |
| def EquivalentNorm (N : Seminorm ℝ X) : Prop := |
| ∃ a b : ℝ, 0 < a ∧ 0 < b ∧ ∀ x, a * ‖x‖ ≤ N x ∧ N x ≤ b * ‖x‖ |
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| def AUC (N : Seminorm ℝ X) : Prop := |
| ∀ t : ℝ, 0 < t → 0 < oneSidedModulus N t |
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| / |
| def Daugavet (X : Type uX) [NormedAddCommGroup X] [NormedSpace ℝ X] : Prop := |
| ∀ (ell : X →L[ℝ] ℝ) (v : X), |
| ‖ContinuousLinearMap.id ℝ X + ell.smulRight v‖ = 1 + ‖ell.smulRight v‖ |
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| variable {Ω : Type u_1} [MeasurableSpace Ω] (μ : Measure Ω) |
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| / |
| def measureDistance (f g : Lp ℝ 1 μ) : ℝ := |
| sInf {a : ℝ | 0 < a ∧ μ.real {ω | a < |f ω - g ω|} < a} |
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| / |
| def MeasurePrecompactBall (E : Submodule ℝ (Lp ℝ 1 μ)) : Prop := |
| ∀ ε : ℝ, 0 < ε → ∃ s : Finset (Lp ℝ 1 μ), |
| ∀ y : E, ‖y‖ ≤ 1 → ∃ b ∈ s, measureDistance μ (y : Lp ℝ 1 μ) b < ε |
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| / |
| def UnitSliceProperty (X : Type uX) [NormedAddCommGroup X] [NormedSpace ℝ X] : Prop := |
| ∀ (w : X), ‖w‖ = 1 → ∀ (ell : X →L[ℝ] ℝ) (α : ℝ), |
| (∃ u : X, ‖u‖ ≤ 1 ∧ α < ell u) → |
| ∀ ρ : ℝ, 0 < ρ → ∃ y : X, ‖y‖ = 1 ∧ α < ell y ∧ 2 - ρ ≤ ‖w + y‖ |
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| / |
| abbrev KWOmega := ℕ → unitInterval |
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| def kwMeasure : Measure KWOmega := Measure.infinitePi (fun _ : ℕ => (volume : Measure unitInterval)) |
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| end ExactModuli |
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| end |
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| end OAI |
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