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0789dda | 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 | 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
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
end OAI
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