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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 | import OAI.Geometry.ProjectionBody.FacetMeasure
import OAI.Geometry.ProjectionBody.Arithmetic
import Mathlib.Basic.ENNReal.BigOperators
namespace OAI
noncomputable section
open Set MeasureTheory
namespace ProjectionCounterexample
variable {ι V W : Type*} [Fintype ι]
[AddCommGroup V] [Module ℝ V] [AddCommGroup W] [Module ℝ W]
[MeasurableSpace W]
/-- Convert the genuine facet-measure sum to the balanced absolute-slope formula.
Every geometric input, including each projected facet measure, remains explicit. -/
theorem real_measure_image_halfspaceBody
(a : ι → V →ₗ[ℝ] ℝ) (b : ι → ℝ) (q : V →ₗ[ℝ] W) (μ : Measure W)
{u : V} (hqu : q u = 0) (hu : ∃ i, 0 < a i u)
(hker : LinearMap.ker q = Submodule.span ℝ {u})
(hmeas : ∀ i, MeasurableSet (q '' boundingFace a b i))
(hnull : ∀ i j, i ≠ j →
μ (q '' (boundingFace a b i ∩ boundingFace a b j)) = 0)
(c : ℝ) (hc : 0 ≤ c) (hbalance : ∑ i, a i u = 0)
(hfacet : ∀ i, μ (q '' boundingFace a b i) = ENNReal.ofReal (c * |a i u|)) :
(μ (q '' halfspaceBody a b)).toReal = c * (∑ i, |a i u|) / 2 := by
classical
rw [measure_image_halfspaceBody_eq_sum_visible a b q μ hqu hu hker hmeas hnull]
simp_rw [hfacet]
rw [ENNReal.toReal_sum (fun _ _ => ENNReal.ofReal_ne_top)]
simp_rw [ENNReal.toReal_ofReal (mul_nonneg hc (abs_nonneg _))]
have hs : (∑ i : {i : ι // 0 < a i u}, |a i.1 u|) =
(∑ i with 0 < a i u, a i u) := by
calc
(∑ i : {i : ι // 0 < a i u}, |a i.1 u|) =
∑ i : {i : ι // 0 < a i u}, a i.1 u := by
apply Finset.sum_congr rfl
intro i _
exact abs_of_pos i.2
_ = _ := (Finset.sum_subtype _ (by simp) (fun i => a i u)).symm
rw [← Finset.mul_sum, hs, sum_positive_eq_half_sum_abs _ hbalance, mul_div_assoc]
end ProjectionCounterexample
end
end OAI
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