| 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 |
| |
