File size: 1,491 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 | import Mathlib.MeasureTheory.Measure.Haar.InnerProductSpace
import OAI.Geometry.ProjectionVolume.Extrusion
namespace OAI
universe uι
open Set MeasureTheory
open scoped Pointwise
namespace Paper092
theorem volume_euclidean_image {ι : Type uι} [Fintype ι] (s : Set (ι → ℝ)) :
(volume : Measure (EuclideanSpace ℝ ι)) (WithLp.toLp 2 '' s) = volume s := by
change volume ((MeasurableEquiv.toLp 2 (ι → ℝ)) '' s) = volume s
rw [MeasurableEquiv.image_eq_preimage_symm]
exact (EuclideanSpace.volume_preserving_symm_measurableEquiv_toLp ι).measure_preimage_equiv s
theorem toLp_image_add_segment {ι : Type uι} [Fintype ι] (s : Set (ι → ℝ)) :
WithLp.toLp 2 '' (s + segment ℝ (0 : ι → ℝ) 1) =
WithLp.toLp 2 '' s + segment ℝ (0 : EuclideanSpace ℝ ι)
(WithLp.toLp 2 (1 : ι → ℝ)) := by
let e := (WithLp.linearEquiv 2 ℝ (ι → ℝ)).symm
have ha := Set.image_add (s := s) (t := segment ℝ (0 : ι → ℝ) 1) e.toAddEquiv
have hs := image_segment ℝ e.toAffineEquiv.toAffineMap (0 : ι → ℝ) 1
simpa [e] using ha.trans (congrArg (fun t => e '' s + t) hs)
theorem volume_euclidean_cube_add_diagonal (d : ℕ) (hd : 0 < d) :
volume ((WithLp.toLp 2 '' Icc (0 : Fin d → ℝ) 1) +
segment ℝ (0 : EuclideanSpace ℝ (Fin d)) (WithLp.toLp 2 (1 : Fin d → ℝ))) =
d + 1 := by
rw [← toLp_image_add_segment, volume_euclidean_image]
exact volume_Icc_add_segment_one d hd
end Paper092
end OAI
|