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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 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 | import OAI.Geometry.ProjectionVolume.SequenceLimits
namespace OAI
open Set Metric Filter
open scoped Pointwise Topology RealInnerProductSpace
namespace Paper092
theorem approximation_linear_image_brightness_uniform {d : ℕ} (hd : 2 ≤ d)
{P : ℕ → Set (Euclidean d)} {K : Set (Euclidean d)}
(hP : ∀ m, IsCompact (P m)) (hK : IsCompact K)
{scale : ℕ → ℝ} (hscale : ∀ m, 1 ≤ scale m) (hscalelim : Tendsto scale atTop (𝓝 1))
(hPK : ∀ m, P m ⊆ K) (hKP : ∀ m, K ⊆ scale m • P m)
(L : Euclidean d ≃ₗ[ℝ] Euclidean d) :
TendstoUniformlyOn (fun m u => brightness (L '' P m) u) (brightness (L '' K))
atTop (sphere 0 1) := by
have hLK : IsCompact (L '' K) := hK.image L.toLinearMap.continuous_of_finiteDimensional
obtain ⟨R, hR, hbound⟩ := hLK.isBounded.exists_pos_norm_le
have hball : L '' K ⊆ closedBall 0 R := by
intro x hx
simpa only [mem_closedBall, dist_zero_right] using hbound x hx
apply approximation_brightness_uniform hd
(fun m => (hP m).image L.toLinearMap.continuous_of_finiteDimensional)
hLK hscale hscalelim hR.le hball
· exact fun m => image_mono (hPK m)
· intro m
rw [← image_smul_comm L.toLinearMap (scale m) (P m)
(fun x => L.toLinearMap.map_smul (scale m) x)]
exact image_mono (hKP m)
theorem approximation_linear_image_support_uniform {d : ℕ} (hd : 2 ≤ d)
{P : ℕ → Set (Euclidean d)} {K : Set (Euclidean d)}
(hP : ∀ m, IsCompact (P m)) (hv : ∀ m, Convex ℝ (P m))
(hi : ∀ m, (interior (P m)).Nonempty) (hK : IsCompact K)
{scale : ℕ → ℝ} (hscale : ∀ m, 1 ≤ scale m) (hscalelim : Tendsto scale atTop (𝓝 1))
(hPK : ∀ m, P m ⊆ K) (hKP : ∀ m, K ⊆ scale m • P m)
(L : Euclidean d ≃ₗ[ℝ] Euclidean d) :
TendstoUniformlyOn
(fun m u => SupportGeometry.support (projectionBody (L '' P m)) (innerSL ℝ u))
(brightness (L '' K)) atTop (sphere 0 1) := by
have hint : ∀ m, (interior (L '' P m)).Nonempty := by
intro m
change (interior (L.toContinuousLinearEquiv.toHomeomorph '' P m)).Nonempty
rw [← Homeomorph.image_interior]
exact (hi m).image _
have heq :
(fun m u => SupportGeometry.support (projectionBody (L '' P m)) (innerSL ℝ u)) =
(fun m u => brightness (L '' P m) u) := by
funext m u
exact projectionBody_support hd (L '' P m)
((hP m).image L.toLinearMap.continuous_of_finiteDimensional)
((hv m).linear_image L.toLinearMap) (hint m) u
rw [heq]
exact approximation_linear_image_brightness_uniform hd hP hK hscale hscalelim hPK hKP L
end Paper092
end OAI
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