File size: 3,324 Bytes
2fca478 | 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 | import OAI.Geometry.ProjectionBody.ProductFacetProjection
import OAI.Geometry.ProjectionBody.SimplexBrightness
import OAI.Geometry.ProjectionBody.ProductVolumes
namespace OAI
noncomputable section
open Set MeasureTheory Measure
namespace ProjectionCounterexample
theorem product_projected_facet_volume (i : Bool × Option (Fin 10))
{u : E 20} (hu : ‖u‖ = 1) :
μHE[19] (project u '' boundingFace productFunctional productLevel i) =
ENNReal.ofReal ((1 / ((Nat.factorial 9 : ℝ) * Nat.factorial 10)) *
|productFunctional i u|) := by
rw [product_facet_eq_affine_image]
have h := measure_projected_affine_chart (productFacetLinear i)
(productFacetCenter i) (project u).toLinearMap (productFacetDomain i.1)
simp only [finrank_euclideanSpace_fin, ContinuousLinearMap.coe_coe] at h
rw [h, productFacetProjection_normDet i hu, volume_productFacetDomain,
← ENNReal.ofReal_mul (abs_nonneg _), mul_comm]
theorem product_projected_facet_measurable
(i : Bool × Option (Fin 10)) (u : E 20) :
MeasurableSet (project u '' boundingFace productFunctional productLevel i) := by
rw [product_facet_eq_affine_image]
apply IsCompact.measurableSet
apply IsCompact.image _ (project u).continuous
apply (productFacetDomain_compact i.1).image
exact continuous_const.add (productFacetLinear i).continuous_of_finiteDimensional
theorem product_projected_facet_intersection_null
(i j : Bool × Option (Fin 10)) (hij : i ≠ j) (u : E 20) :
μHE[19] (project u ''
(boundingFace productFunctional productLevel i ∩
boundingFace productFunctional productLevel j)) = 0 := by
have hS : boundingFace productFunctional productLevel i ⊆
(fun x => productFacetCenter i + productFacetLinear i x) '' (univ : Set (E 19)) := by
rw [product_facet_eq_affine_image]
exact Set.image_mono (subset_univ _)
simpa only [finrank_euclideanSpace_fin, ContinuousLinearMap.coe_coe] using
measure_projected_transverse_intersection (productFacetLinear i)
(productFacetCenter i) (project u).toLinearMap (productFunctional j) (productLevel j)
_ _ hS (fun _ h => h.2) (product_facet_transverse i j hij)
/-- Actual orthogonal projection volume of the twenty-dimensional product body. -/
theorem product_projectionVolume {u : E 20} (hu : ‖u‖ = 1) :
projectionVolume productBody u =
(((∑ i, |firstBlock u i|) + |∑ i, firstBlock u i|) +
((∑ i, |secondBlock u i|) + |∑ i, secondBlock u i|)) /
(2 * ((Nat.factorial 9 : ℝ) * Nat.factorial 10)) := by
have hu0 : u ≠ 0 := by intro h; simp [h] at hu
have h := real_measure_image_halfspaceBody
productFunctional productLevel (project u).toLinearMap μHE[19]
(project_self u) (product_has_positive_slope hu0) (project_ker u)
(fun i => product_projected_facet_measurable i u)
(fun i j hij => product_projected_facet_intersection_null i j hij u)
(1 / ((Nat.factorial 9 : ℝ) * Nat.factorial 10)) (by positivity)
(product_functional_balance u) (fun i => product_projected_facet_volume i hu)
rw [← productBody_eq_halfspaceBody, product_absolute_slopes] at h
simp only [ContinuousLinearMap.coe_coe] at h
unfold projectionVolume
change (μHE[19] (project u '' productBody)).toReal = _
rw [h]
ring
end ProjectionCounterexample
end
end OAI
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