File size: 2,525 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 | 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
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