| import OAI.Geometry.CylinderCovering.Caps.FiniteDefinitions |
| import OAI.Geometry.CylinderCovering.Radial.Riemann |
| import OAI.Geometry.CylinderCovering.Caps.FiniteArea |
|
|
| namespace OAI |
|
|
| noncomputable section |
| namespace CylinderCovering.HalfheightFinite |
| open Set Filter MeasureTheory |
| open scoped Topology |
|
|
| lemma midpoint_isTagged (N : ℕ) : |
| RadialRiemann.IsTagged (-1) 1 N (midpoint N) := by |
| intro i |
| have hd := RadialRiemann.step_pos (by norm_num : (-1 : ℝ) < 1) |
| (Nat.zero_lt_of_lt i.isLt) |
| have hs := RadialRiemann.point_succ (-1) 1 N i |
| change RadialRiemann.point (-1) 1 N i ≤ midpoint N i ∧ |
| midpoint N i ≤ RadialRiemann.point (-1) 1 N (i+1) |
| dsimp [midpoint] |
| constructor <;> linarith |
|
|
| lemma projectionWeight_continuous (δ : ℝ) : |
| Continuous (fun r : ℝ => |
| 1 / Real.sqrt (1 + δ^2 * (alpha r^2 + 2 * beta r^2))) := by |
| apply continuous_const.div |
| (show Continuous (fun r : ℝ => |
| Real.sqrt (1 + δ^2 * (alpha r^2 + 2 * beta r^2))) by |
| unfold alpha beta |
| fun_prop) |
| intro r |
| apply ne_of_gt |
| apply Real.sqrt_pos.mpr |
| positivity |
|
|
| theorem localArea_sum_tendsto (δ : ℝ) {R : ℝ → ℝ} |
| (hR : ContinuousOn R (Icc (-1 : ℝ) 1)) : |
| Tendsto (fun N => ∑ i : Fin N, localArea δ R N i) atTop |
| (𝓝 (limitingArea δ R / 2)) := by |
| have h := RadialRiemann.maxPolynomialSum_tendsto |
| (by norm_num : (-1 : ℝ) < 1) hR |
| (projectionWeight_continuous δ).continuousOn |
| (height/4) (3*height*δ/8) (height*δ^2/8) |
| midpoint midpoint_isTagged |
| convert h using 1 |
| · funext N |
| apply Finset.sum_congr rfl |
| intro i _ |
| simp only [localArea, maxCap, collar, halfwidth, RadialRiemann.step] |
| ring |
| · unfold limitingArea |
| have hi : (∫ r in (-1 : ℝ)..1, |
| R r^2 / Real.sqrt (1+δ^2*(alpha r^2+2*beta r^2))) = |
| ∫ r in (-1 : ℝ)..1, |
| R r^2 * (1 / Real.sqrt (1+δ^2*(alpha r^2+2*beta r^2))) := by |
| apply intervalIntegral.integral_congr |
| intro r _ |
| ring |
| rw [hi] |
| congr 1 |
| ring |
|
|
| theorem totalArea_tendsto {δ : ℝ} (hδ : 0 ≤ δ) {R : ℝ → ℝ} |
| (hR : ContinuousOn R (Icc (-1 : ℝ) 1)) |
| (hR0 : ∀ r ∈ Icc (-1 : ℝ) 1, 0 ≤ R r) : |
| Tendsto (totalArea δ R) atTop (𝓝 (limitingArea δ R)) := by |
| have h := (localArea_sum_tendsto δ hR).const_mul 2 |
| simpa only [← totalArea_eq_two_sum hδ hR hR0, mul_div_cancel₀ _ (by norm_num : (2 : ℝ) ≠ 0)] using h |
|
|
| end CylinderCovering.HalfheightFinite |
|
|
| end |
|
|
| end OAI |
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|
