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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 | import OAI.Geometry.ProjectionVolume.Basic
namespace OAI
universe uι
open Set MeasureTheory
open scoped RealInnerProductSpace Pointwise
noncomputable section
namespace Paper092
structure HPolytope (d : ℕ) (ι : Type uι) [Fintype ι] where
normal : ι → Euclidean d
offset : ι → ℝ
normal_unit : ∀ i, ‖normal i‖ = 1
compact : IsCompact {x : Euclidean d | ∀ i, ⟪normal i, x⟫ ≤ offset i}
strict_feasible : ∃ x : Euclidean d, ∀ i, ⟪normal i, x⟫ < offset i
distinct : Function.Injective (fun i => (normal i, offset i))
namespace HPolytope
def body {d : ℕ} {ι : Type uι} [Fintype ι] (P : HPolytope d ι) : Set (Euclidean d) :=
{x | ∀ i, ⟪P.normal i, x⟫ ≤ P.offset i}
def face {d : ℕ} {ι : Type uι} [Fintype ι] (P : HPolytope d ι) (i : ι) :
Set (Euclidean d) :=
{x | x ∈ P.body ∧ ⟪P.normal i, x⟫ = P.offset i}
def faceArea {d : ℕ} {ι : Type uι} [Fintype ι] (P : HPolytope d ι) (i : ι) : ℝ :=
(μHE[d - 1] (P.face i)).toReal
def facetZonotope {d : ℕ} {ι : Type uι} [Fintype ι] (P : HPolytope d ι) :
Set (Euclidean d) :=
∑ i, segment ℝ (-(P.faceArea i / 2) • P.normal i) ((P.faceArea i / 2) • P.normal i)
end HPolytope
end Paper092
end
end OAI
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