| 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) |
|
|
| / |
| 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 |
|
|
