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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 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 | import OAI.Geometry.CylinderCovering.Simplex.ProjectionDefinitions
import OAI.Geometry.CylinderCovering.Simplex.Diameter
namespace OAI
open Set Metric MeasureTheory
namespace CylinderCovering
lemma longestChord_eq_maxChord {d : ℕ} (S : Set (Euclidean d)) (u : Euclidean d) :
longestChord S u = SimplexDiameter.maxChord S u := rfl
lemma simplexMinimumShadowArea_of_product {d : ℕ} (hd : 2 ≤ d)
(v : Fin (d + 1) → Euclidean d) (hv : AffineIndependent ℝ v)
(hprod : ∀ u, ‖u‖ = 1 → simplexShadowArea v u * longestChord (simplexCarrier v) u =
(d : ℝ) * volume.real (simplexCarrier v)) :
simplexMinimumShadowArea v =
(d : ℝ) * volume.real (simplexCarrier v) / diam (simplexCarrier v) := by
have : Nontrivial (Fin (d + 1)) := Fin.nontrivial_iff_two_le.mpr (by omega)
have hleast := SimplexDiameter.projection_minimum_of_chord_identity
((finite_range v).isCompact_convexHull ℝ)
((range_nonempty v).mono (subset_convexHull ℝ _))
(SimplexDiameter.affineIndependent_diameter_pos hv)
(simplexShadowArea v) ((d : ℝ) * volume.real (simplexCarrier v))
(fun _ _ => ENNReal.toReal_nonneg) hprod
unfold simplexMinimumShadowArea
convert hleast.csInf_eq using 1 <;> congr 1
ext a
simp only [eq_comm]
lemma tetrahedronEdgeMinimizer_of_simplexChordProjection
(h : SimplexChordProjectionStatement) : TetrahedronEdgeMinimizerStatement := by
intro v hv
obtain ⟨hdir, _, hmin⟩ := h 3 (by omega) v hv
have hprod := fun u hu => (hdir u hu).2.2
obtain ⟨i, j, hij, hu, hval, _⟩ := SimplexDiameter.finite_edge_projection_minimizer v
(SimplexDiameter.affineIndependent_diameter_pos hv)
(simplexShadowArea v) ((3 : ℝ) * volume.real (simplexCarrier v))
(fun _ _ => ENNReal.toReal_nonneg) hprod
refine ⟨i, j, hij, ?_⟩
change simplexShadowArea v (NormedSpace.normalize (v j - v i)) =
simplexMinimumShadowArea v
exact hval.trans hmin.symm
lemma simplexCarrier_isCompact {d : ℕ} (v : Fin (d + 1) → Euclidean d) :
IsCompact (simplexCarrier v) := (finite_range v).isCompact_convexHull ℝ
lemma simplexCarrier_interior_nonempty {d : ℕ}
(v : Fin (d + 1) → Euclidean d) (hv : AffineIndependent ℝ v) :
(interior (simplexCarrier v)).Nonempty := by
apply (convex_convexHull ℝ (range v)).interior_nonempty_iff_affineSpan_eq_top.mpr
rw [affineSpan_convexHull]
apply hv.affineSpan_eq_top_iff_card_eq_finrank_add_one.mpr
simp
lemma simplexCarrier_volume_pos {d : ℕ}
(v : Fin (d + 1) → Euclidean d) (hv : AffineIndependent ℝ v) :
0 < volume.real (simplexCarrier v) := by
apply ENNReal.toReal_pos
· exact (Measure.measure_pos_of_nonempty_interior volume
(simplexCarrier_interior_nonempty v hv)).ne'
· exact (simplexCarrier_isCompact v).measure_ne_top
lemma simplex_longestChord_pos {d : ℕ}
(v : Fin (d + 1) → Euclidean d) (hv : AffineIndependent ℝ v)
{u : Euclidean d} (hu : ‖u‖ = 1) :
0 < longestChord (simplexCarrier v) u :=
SimplexDiameter.maxChord_pos_of_interior (simplexCarrier_isCompact v)
(simplexCarrier_interior_nonempty v hv) hu
lemma simplexMinimumShadowArea_attained_of_product {d : ℕ} (hd : 2 ≤ d)
(v : Fin (d + 1) → Euclidean d) (hv : AffineIndependent ℝ v)
(hprod : ∀ u, ‖u‖ = 1 → simplexShadowArea v u * longestChord (simplexCarrier v) u =
(d : ℝ) * volume.real (simplexCarrier v)) :
∃ u, ‖u‖ = 1 ∧ simplexShadowArea v u = simplexMinimumShadowArea v := by
have : Nontrivial (Fin (d + 1)) := Fin.nontrivial_iff_two_le.mpr (by omega)
obtain ⟨u, hu, hmax⟩ := SimplexDiameter.exists_maxChord_eq_diam
(simplexCarrier_isCompact v) ((range_nonempty v).mono (subset_convexHull ℝ _))
(SimplexDiameter.affineIndependent_diameter_pos hv)
simp only [simplexCarrier] at hmax
refine ⟨u, hu, ?_⟩
rw [simplexMinimumShadowArea_of_product hd v hv hprod]
apply (eq_div_iff (SimplexDiameter.affineIndependent_diameter_pos hv).ne').mpr
simpa only [longestChord_eq_maxChord, simplexCarrier, hmax] using hprod u hu
lemma simplexChordProjection_of_products
(hprod : ∀ d : ℕ, 2 ≤ d → ∀ v : Fin (d + 1) → Euclidean d,
AffineIndependent ℝ v → ∀ u, ‖u‖ = 1 →
simplexShadowArea v u * longestChord (simplexCarrier v) u =
(d : ℝ) * volume.real (simplexCarrier v)) :
SimplexChordProjectionStatement := by
intro d hd v hv
refine ⟨?_, simplexMinimumShadowArea_attained_of_product hd v hv (hprod d hd v hv),
simplexMinimumShadowArea_of_product hd v hv (hprod d hd v hv)⟩
intro u hu
exact ⟨simplex_longestChord_pos v hv hu,
(SimplexDiameter.maxChord_attained (simplexCarrier_isCompact v)
((range_nonempty v).mono (subset_convexHull ℝ _)) hu).2,
hprod d hd v hv u hu⟩
lemma regular_edge_shadow_eq_minimum_of_simplexChordProjection
(h : SimplexChordProjectionStatement)
(v : Fin 4 → Space) (hv : AffineIndependent ℝ v)
{edge : ℝ} (he : 0 < edge)
(hvEdge : ∀ i j : Fin 4, i ≠ j → dist (v i) (v j) = edge)
{i j : Fin 4} (hij : i ≠ j) :
shadowArea (convexHull ℝ (range v)) (‖v j - v i‖⁻¹ • (v j - v i)) =
minimumShadowArea (convexHull ℝ (range v)) := by
obtain ⟨hu, hchord⟩ := SimplexDiameter.regular_edge_maxChord v he hvEdge hij
obtain ⟨hdir, _, hmin⟩ := h 3 (by omega) v hv
change simplexShadowArea v (NormedSpace.normalize (v j - v i)) =
simplexMinimumShadowArea v
have hprod := (hdir _ hu).2.2
have hd : diam (simplexCarrier v) = edge :=
SimplexDiameter.regular_convexHull_diameter v he.le hvEdge
rw [hmin, hd]
apply (eq_div_iff he.ne').mpr
simpa only [longestChord_eq_maxChord, simplexCarrier, hchord] using hprod
lemma regularProjectionNormalization_of_minimum
(h : SimplexChordProjectionStatement)
(hmin : ∀ 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)) :
RegularProjectionNormalizationStatement := by
intro v hv edge he hvEdge
refine ⟨hmin v hv edge he hvEdge, ?_⟩
intro i j hij
exact (regular_edge_shadow_eq_minimum_of_simplexChordProjection h v hv he hvEdge hij).trans
(hmin v hv edge he hvEdge)
end CylinderCovering
end OAI
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