File size: 4,023 Bytes
d5b7741 | 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 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 | import Mathlib.MeasureTheory.Integral.Bochner.Set
import Mathlib.MeasureTheory.Measure.Lebesgue.Basic
import Mathlib.MeasureTheory.Constructions.Pi
import Mathlib.Analysis.SpecialFunctions.Pow.Real
namespace OAI
/-!
# The published analytic input for result 200
Prékopa, *On logarithmic concave measures and functions*, Acta Sci. Math.
(Szeged) 34 (1973), 335–343, Theorem 3, equation (2.3), with two source
functions. Source: https://acta.bibl.u-szeged.hu/14411/1/math_034_335-343.pdf
The one-dimensional input is Leindler (1972), *On a certain converse of
Hölder's inequality II*, 217–223, as identified in Prékopa's §2.
The finite-box specialization allows positive continuous functions on a
common compact coordinate box and finite unions of open boxes.
-/
namespace ContingencyTables.PublishedInputs
open MeasureTheory
abbrev Point (q : ℕ) := Fin q → ℝ
/-- The bounded finite unions of open coordinate boxes used by the bin
conductance argument. Empty unions and empty coordinate boxes are allowed. -/
def IsFiniteBoxUnion {q : ℕ} (S : Set (Point q)) : Prop :=
∃ (n : ℕ) (lower upper : Fin n → Point q),
S = {x | ∃ k : Fin n, ∀ i, lower k i < x i ∧ x i < upper k i}
/-- The weighted Minkowski interpolation with the manuscript convention. -/
def interpolate {q : ℕ} (t : ℝ) (S T : Set (Point q)) : Set (Point q) :=
{z | ∃ x ∈ S, ∃ y ∈ T, z = (1 - t) • x + t • y}
/-- Precisely the previously published finite-box integral interpolation
input. The functions need only be positive and continuous on the common
compact coordinate box; all integrals are Lebesgue integrals. -/
def FiniteBoxPrekopaLeindler (q : ℕ) : Prop :=
∀ (t : ℝ), 0 < t → t < 1 →
∀ (lower upper : Point q) (S₀ S₁ S₂ : Set (Point q))
(f₀ f₁ f₂ : Point q → ℝ),
IsFiniteBoxUnion S₀ → IsFiniteBoxUnion S₁ → IsFiniteBoxUnion S₂ →
closure S₀ ⊆ Set.Icc lower upper →
closure S₁ ⊆ Set.Icc lower upper →
closure S₂ ⊆ Set.Icc lower upper →
ContinuousOn f₀ (Set.Icc lower upper) →
ContinuousOn f₁ (Set.Icc lower upper) →
ContinuousOn f₂ (Set.Icc lower upper) →
(∀ x ∈ Set.Icc lower upper, 0 < f₀ x) →
(∀ x ∈ Set.Icc lower upper, 0 < f₁ x) →
(∀ x ∈ Set.Icc lower upper, 0 < f₂ x) →
interpolate t S₁ S₂ ⊆ S₀ →
(∀ x ∈ S₁, ∀ y ∈ S₂,
(f₁ x) ^ (1 - t) * (f₂ y) ^ t ≤ f₀ ((1 - t) • x + t • y)) →
(∫ x in S₁, f₁ x) ^ (1 - t) * (∫ y in S₂, f₂ y) ^ t ≤
∫ z in S₀, f₀ z
/-- Log-concavity is needed only on the common ambient box. -/
def LogConcaveOn {q : ℕ} (ρ : Point q → ℝ) (K : Set (Point q)) : Prop :=
∀ x ∈ K, ∀ y ∈ K, ∀ t : ℝ, 0 < t → t < 1 →
(ρ x) ^ (1 - t) * (ρ y) ^ t ≤ ρ ((1 - t) • x + t • y)
/-- Specialization of the published input to the one log-concave density
used for bin conductance. The geometric hypotheses specify the common ambient box. -/
theorem density_interpolation {q : ℕ} (hPL : FiniteBoxPrekopaLeindler q)
(t : ℝ) (ht : 0 < t) (ht1 : t < 1)
(lower upper : Point q) (S T : Set (Point q)) (ρ : Point q → ℝ)
(hS : IsFiniteBoxUnion S) (hT : IsFiniteBoxUnion T)
(hM : IsFiniteBoxUnion (interpolate t S T))
(hSc : closure S ⊆ Set.Icc lower upper)
(hTc : closure T ⊆ Set.Icc lower upper)
(hMc : closure (interpolate t S T) ⊆ Set.Icc lower upper)
(hρ : ContinuousOn ρ (Set.Icc lower upper))
(hpos : ∀ x ∈ Set.Icc lower upper, 0 < ρ x)
(hlc : LogConcaveOn ρ (Set.Icc lower upper)) :
(∫ x in S, ρ x) ^ (1 - t) * (∫ y in T, ρ y) ^ t ≤
∫ z in interpolate t S T, ρ z := by
apply hPL t ht ht1 lower upper (interpolate t S T) S T ρ ρ ρ
hM hS hT hMc hSc hTc hρ hρ hρ hpos hpos hpos Set.Subset.rfl
intro x hx y hy
exact hlc x (hSc (subset_closure hx)) y (hTc (subset_closure hy)) t ht ht1
end ContingencyTables.PublishedInputs
end OAI
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