| 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 | |