| import Mathlib |
|
|
| namespace OAI |
|
|
| open MeasureTheory ProbabilityTheory |
| open scoped ENNReal |
|
|
| namespace GaussianPropeller |
|
|
| abbrev Space (d : ℕ) := EuclideanSpace ℝ (Fin d) |
|
|
| noncomputable def gaussian (d : ℕ) : Measure (Space d) := stdGaussian (Space d) |
|
|
| def IsPartition {d k : ℕ} (A : Fin k → Set (Space d)) : Prop := |
| (∀ i, MeasurableSet (A i)) ∧ ∀ᵐ x ∂gaussian d, ∃! i, x ∈ A i |
|
|
| noncomputable def centroid {d : ℕ} (A : Set (Space d)) : Space d := |
| ∫ x in A, x ∂gaussian d |
|
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| noncomputable def value {d k : ℕ} (A : Fin k → Set (Space d)) : ℝ := |
| ∑ i, ‖centroid (A i)‖ ^ 2 |
|
|
| def coord {d : ℕ} (j : ℕ) (x : Space d) : ℝ := |
| if h : j < d then x ⟨j, h⟩ else 0 |
|
|
| noncomputable def propeller (d k : ℕ) (i : Fin k) : Set (Space d) := |
| if i.val = 0 then {x | |coord 1 x| ≤ Real.sqrt 3 * coord 0 x} |
| else if i.val = 1 then {x | 0 ≤ coord 1 x ∧ Real.sqrt 3 * coord 0 x ≤ coord 1 x} |
| else if i.val = 2 then {x | coord 1 x ≤ 0 ∧ Real.sqrt 3 * coord 0 x ≤ -coord 1 x} |
| else ∅ |
|
|
| def AllPartitions : Prop := |
| (∀ (d k : ℕ), 0 < d → 0 < k → ∀ A : Fin k → Set (Space d), |
| IsPartition A → value A ≤ 9 / (8 * Real.pi)) ∧ |
| (∀ (d k : ℕ), 2 ≤ d → 3 ≤ k → |
| IsPartition (propeller d k) ∧ value (propeller d k) = 9 / (8 * Real.pi)) |
|
|
| theorem all_partitions : AllPartitions := by |
| sorry |
|
|
| end GaussianPropeller |
|
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| end OAI |
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|
