File size: 2,649 Bytes
ddc84c5 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 | import OAI.Analysis.Daugavet.EquivalentNorm
namespace OAI
noncomputable section
open MeasureTheory Filter Set
open scoped ENNReal NNReal Topology
universe uX u_1
namespace ExactModuli
variable {X : Type uX} [NormedAddCommGroup X] [NormedSpace ℝ X]
/-- 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
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)⟩
end ExactModuli
end
end OAI
|