| import OAI.Geometry.ProjectionBody.Definitions | |
| import OAI.Geometry.ProjectionBody.UnitSupport | |
| import OAI.Geometry.Zonotope.Euclidean | |
| namespace OAI | |
| /-! | |
| Generic bridge from a *proved* directional projection-volume formula to the | |
| explicit zonotope. The brightness formula is an explicit hypothesis of these | |
| helper lemmas; no final counterexample theorem assumes it. | |
| -/ | |
| noncomputable section | |
| open Set | |
| open scoped RealInnerProductSpace | |
| namespace ProjectionCounterexample | |
| /-- A unit-direction brightness formula identifies the actual projection body. -/ | |
| theorem projectionBody_eq_zonotope {d : ℕ} (P : Set (E d)) (c : ℝ) (hc : 0 < c) | |
| (hbrightness : ∀ u : E d, ‖u‖ = 1 → | |
| projectionVolume P u = c * DiagonalZonotope.support (WithLp.ofLp u)) : | |
| projectionBody P = DiagonalZonotope.euclidean c := by | |
| calc | |
| projectionBody P = {x : E d | ∀ u : E d, ‖u‖ = 1 → | |
| ⟪u, x⟫ ≤ c * DiagonalZonotope.support (WithLp.ofLp u)} := by | |
| ext x | |
| constructor | |
| · intro hx u hu | |
| rw [← hbrightness u hu] | |
| exact hx u hu | |
| · intro hx u hu | |
| rw [hbrightness u hu] | |
| exact hx u hu | |
| _ = {x : E d | ∀ u : E d, | |
| ⟪u, x⟫ ≤ c * DiagonalZonotope.support (WithLp.ofLp u)} := by | |
| apply unit_support_halfspaces_eq | |
| · simp [DiagonalZonotope.support] | |
| · intro a ha u | |
| change c * DiagonalZonotope.support (a • WithLp.ofLp u) = _ | |
| rw [DiagonalZonotope.support_smul, abs_of_nonneg ha] | |
| ring | |
| _ = DiagonalZonotope.euclidean c := | |
| (DiagonalZonotope.euclidean_eq_halfspaces c hc).symm | |
| /-- The brightness hypothesis gives both a genuine body and support attainment. -/ | |
| theorem projection_zonotope_is_body_and_attains {d : ℕ} (P : Set (E d)) | |
| (c : ℝ) (hc : 0 < c) | |
| (hbrightness : ∀ u : E d, ‖u‖ = 1 → | |
| projectionVolume P u = c * DiagonalZonotope.support (WithLp.ofLp u)) : | |
| IsConvexBody (projectionBody P) ∧ | |
| ∀ u : E d, ‖u‖ = 1 → ∃ x ∈ projectionBody P, ⟪u, x⟫ = projectionVolume P u := by | |
| have hbody := projectionBody_eq_zonotope P c hc hbrightness | |
| constructor | |
| · rw [hbody] | |
| exact ⟨DiagonalZonotope.isCompact_euclidean c, | |
| DiagonalZonotope.convex_euclidean c, | |
| DiagonalZonotope.euclidean_interior_nonempty c hc.ne'⟩ | |
| · intro u hu | |
| obtain ⟨x, hx, heq⟩ := DiagonalZonotope.euclidean_support_attained c u | |
| refine ⟨x, hbody.symm ▸ hx, ?_⟩ | |
| rw [hbrightness u hu] | |
| exact heq | |
| end ProjectionCounterexample | |
| end | |
| end OAI | |