File size: 2,590 Bytes
785c727 | 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 | import OAI.Geometry.CylinderCovering.Convexity.GlobalDefinitions
namespace OAI
noncomputable section
open Set MeasureTheory
open scoped InnerProductSpace
namespace CylinderCovering
abbrev Euclidean (d : ℕ) := EuclideanSpace ℝ (Fin d)
def simplexCarrier {d : ℕ} (v : Fin (d + 1) → Euclidean d) : Set (Euclidean d) :=
convexHull ℝ (range v)
def orthogonalProjection {d : ℕ} (u x : Euclidean d) : Euclidean d :=
x - inner ℝ u x • u
def simplexShadowArea {d : ℕ} (v : Fin (d + 1) → Euclidean d)
(u : Euclidean d) : ℝ :=
(Measure.euclideanHausdorffMeasure (d - 1)
(orthogonalProjection u '' simplexCarrier v)).toReal
def longestChord {d : ℕ} (S : Set (Euclidean d)) (u : Euclidean d) : ℝ :=
sSup {r : ℝ | 0 ≤ r ∧ ∃ x ∈ S, x + r • u ∈ S}
def simplexMinimumShadowArea {d : ℕ} (v : Fin (d + 1) → Euclidean d) : ℝ :=
sInf {a : ℝ | ∃ u : Euclidean d, ‖u‖ = 1 ∧ a = simplexShadowArea v u}
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)
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))
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)
def SimplexProjectionIdentityStatement : Prop :=
SimplexChordProjectionStatement ∧ TetrahedronEdgeMinimizerStatement ∧
RegularProjectionNormalizationStatement
end CylinderCovering
end
end OAI
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