| import OAI.Geometry.CylinderCovering.Convexity.GlobalDefinitions |
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| namespace OAI |
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| noncomputable section |
| open Set MeasureTheory |
| open scoped InnerProductSpace |
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| namespace CylinderCovering |
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| abbrev Euclidean (d : ℕ) := EuclideanSpace ℝ (Fin d) |
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| def simplexCarrier {d : ℕ} (v : Fin (d + 1) → Euclidean d) : Set (Euclidean d) := |
| convexHull ℝ (range v) |
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| def orthogonalProjection {d : ℕ} (u x : Euclidean d) : Euclidean d := |
| x - inner ℝ u x • u |
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| def simplexShadowArea {d : ℕ} (v : Fin (d + 1) → Euclidean d) |
| (u : Euclidean d) : ℝ := |
| (Measure.euclideanHausdorffMeasure (d - 1) |
| (orthogonalProjection u '' simplexCarrier v)).toReal |
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| def longestChord {d : ℕ} (S : Set (Euclidean d)) (u : Euclidean d) : ℝ := |
| sSup {r : ℝ | 0 ≤ r ∧ ∃ x ∈ S, x + r • u ∈ S} |
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| def simplexMinimumShadowArea {d : ℕ} (v : Fin (d + 1) → Euclidean d) : ℝ := |
| sInf {a : ℝ | ∃ u : Euclidean d, ‖u‖ = 1 ∧ a = simplexShadowArea v u} |
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| def SimplexChordProjectionStatement : Prop := |
| ∀ d : ℕ, 2 ≤ d → ∀ v : Fin (d + 1) → Euclidean d, |
| AffineIndependent ℝ v → |
| (∀ u : Euclidean d, ‖u‖ = 1 → |
| 0 < longestChord (simplexCarrier v) u ∧ |
| (∃ x ∈ simplexCarrier v, |
| x + longestChord (simplexCarrier v) u • u ∈ simplexCarrier v) ∧ |
| simplexShadowArea v u * longestChord (simplexCarrier v) u = |
| (d : ℝ) * volume.real (simplexCarrier v)) ∧ |
| (∃ u : Euclidean d, ‖u‖ = 1 ∧ |
| simplexShadowArea v u = simplexMinimumShadowArea v) ∧ |
| simplexMinimumShadowArea v = |
| (d : ℝ) * volume.real (simplexCarrier v) / Metric.diam (simplexCarrier v) |
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| def TetrahedronEdgeMinimizerStatement : Prop := |
| ∀ v : Fin 4 → Space, AffineIndependent ℝ v → |
| ∃ i j : Fin 4, i ≠ j ∧ |
| shadowArea (convexHull ℝ (range v)) |
| (‖v j - v i‖⁻¹ • (v j - v i)) = |
| minimumShadowArea (convexHull ℝ (range v)) |
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| def RegularProjectionNormalizationStatement : Prop := |
| ∀ v : Fin 4 → Space, AffineIndependent ℝ v → |
| ∀ edge : ℝ, 0 < edge → |
| (∀ i j : Fin 4, i ≠ j → dist (v i) (v j) = edge) → |
| minimumShadowArea (convexHull ℝ (range v)) = edge ^ 2 / (2 * Real.sqrt 2) ∧ |
| ∀ i j : Fin 4, i ≠ j → |
| shadowArea (convexHull ℝ (range v)) |
| (‖v j - v i‖⁻¹ • (v j - v i)) = edge ^ 2 / (2 * Real.sqrt 2) |
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| def SimplexProjectionIdentityStatement : Prop := |
| SimplexChordProjectionStatement ∧ TetrahedronEdgeMinimizerStatement ∧ |
| RegularProjectionNormalizationStatement |
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| end CylinderCovering |
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| end |
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| end OAI |
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