File size: 4,378 Bytes
cead56f | 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 | import OAI.Geometry.ProjectionBody.SimplexFacetCharts
import OAI.Geometry.ProjectionBody.GraphProjection
import OAI.Geometry.ProjectionBody.SimplexVolume
import OAI.Geometry.ProjectionBody.AffineChartMeasure
import OAI.Geometry.ProjectionBody.BrightnessSum
import OAI.Geometry.ProjectionBody.ProjectionKernel
namespace OAI
noncomputable section
open Set MeasureTheory Measure
open scoped RealInnerProductSpace
namespace ProjectionCounterexample
theorem volume_simplex (n : ℕ) :
volume (simplex n) = ENNReal.ofReal (1 / (n.factorial : ℝ)) := by
rw [simplex_eq_coordinates]
exact volume_unit_euclideanSimplex n
theorem simplexFacetProjection_normDet {n : ℕ} (i : Option (Fin (n + 1)))
{u : E (n + 1)} (hu : ‖u‖ = 1) :
((project u).toLinearMap.comp (simplexFacetLinear n i)).normDet =
|simplexFunctional (n + 1) i u| := by
cases i with
| none =>
change ((project u).toLinearMap.comp
(graphLinear (Fin.last n) (allOnes n))).normDet = _
rw [graphProjection_normDet _ _ hu]
have hnormal : graphNormal (Fin.last n) (allOnes n) = allOnes (n + 1) := by
ext j
induction j using Fin.lastCases <;>
simp [graphNormal, insertZero, Fin.insertNth_last']
simp [hnormal]
| some i =>
change ((project u).toLinearMap.comp (insertZero i).toLinearMap).normDet = _
rw [coordinateProjection_normDet i hu]
simp
theorem simplex_projected_facet_volume {n : ℕ} (i : Option (Fin (n + 1)))
{u : E (n + 1)} (hu : ‖u‖ = 1) :
μHE[n] (project u '' boundingFace (simplexFunctional (n + 1)) simplexLevel i) =
ENNReal.ofReal ((1 / (n.factorial : ℝ)) * |simplexFunctional (n + 1) i u|) := by
rw [simplex_facet_eq_affine_image]
have h := measure_projected_affine_chart (simplexFacetLinear n i)
(simplexFacetCenter n i) (project u).toLinearMap (simplex n)
simp only [finrank_euclideanSpace_fin, ContinuousLinearMap.coe_coe] at h
rw [h, simplexFacetProjection_normDet i hu, volume_simplex,
← ENNReal.ofReal_mul (abs_nonneg _), mul_comm]
theorem simplex_projected_facet_measurable {n : ℕ}
(i : Option (Fin (n + 1))) (u : E (n + 1)) :
MeasurableSet (project u '' boundingFace (simplexFunctional (n + 1)) simplexLevel i) := by
rw [simplex_facet_eq_affine_image]
apply IsCompact.measurableSet
apply IsCompact.image _ (project u).continuous
apply (simplex_compact n).image
exact continuous_const.add (simplexFacetLinear n i).continuous_of_finiteDimensional
theorem simplex_projected_facet_intersection_null {n : ℕ} (hn : 0 < n)
(i j : Option (Fin (n + 1))) (hij : i ≠ j) (u : E (n + 1)) :
μHE[n] (project u ''
(boundingFace (simplexFunctional (n + 1)) simplexLevel i ∩
boundingFace (simplexFunctional (n + 1)) simplexLevel j)) = 0 := by
have hS : boundingFace (simplexFunctional (n + 1)) simplexLevel i ⊆
(fun x => simplexFacetCenter n i + simplexFacetLinear n i x) '' (univ : Set (E n)) := by
rw [simplex_facet_eq_affine_image]
exact Set.image_mono (subset_univ _)
simpa only [finrank_euclideanSpace_fin, ContinuousLinearMap.coe_coe] using
measure_projected_transverse_intersection (simplexFacetLinear n i)
(simplexFacetCenter n i) (project u).toLinearMap
(simplexFunctional (n + 1) j) (simplexLevel j) _ _ hS
(fun _ h => h.2) (simplex_facet_transverse hn i j hij)
/-- Actual orthogonal projection volume of the convex-hull standard simplex. -/
theorem simplex_projectionVolume {n : ℕ} (hn : 0 < n)
{u : E (n + 1)} (hu : ‖u‖ = 1) :
projectionVolume (simplex (n + 1)) u =
((∑ i, |u i|) + |∑ i, u i|) / (2 * (n.factorial : ℝ)) := by
have hu0 : u ≠ 0 := by intro h; simp [h] at hu
have h := real_measure_image_halfspaceBody
(simplexFunctional (n + 1)) simplexLevel (project u).toLinearMap μHE[n]
(project_self u) (simplex_has_positive_slope hu0) (project_ker u)
(fun i => simplex_projected_facet_measurable i u)
(fun i j hij => simplex_projected_facet_intersection_null hn i j hij u)
(1 / (n.factorial : ℝ)) (by positivity) (simplex_functional_balance u)
(fun i => simplex_projected_facet_volume i hu)
rw [← simplex_eq_halfspaceBody, simplex_absolute_slopes] at h
simp only [ContinuousLinearMap.coe_coe] at h
unfold projectionVolume
simp only [Nat.add_sub_cancel]
rw [h]
ring
end ProjectionCounterexample
end
end OAI
|