Inscribed rectangles in a smooth Jordan curve attain at least one third of all aspect ratios

Problem statement

Notes: Unavailable.

Source: C. Hugelmeyer, `Inscribed rectangles in a smooth Jordan curve attain at least one third of all aspect ratios`, Annals of Math, 194 (2) 2021. Statement taken from https://github.com/ImperialCollegeLondon/AnnalsChallenge (v1.0.0, e32eb14), AnnalsChallenge/AnnalsOfMathematics/2021-194-2-InscribedRectangles.lean

Informal solution: Unavailable.

theorem declaration uses `sorry`theorem_1 (γ : Circle → ℝ × ℝ) (hγ : IsSmoothEmbedding (𝓡 1) 𝓘(ℝ, ℝ × ℝ) ∞ γ) :
    volume (X γ) ≥ 1/3 := byγ:Circle → ℝ × ℝhγ:IsSmoothEmbedding (𝓡 1) 𝓘(ℝ, ℝ × ℝ) ∞ γ⊢ volume (X γ) ≥ 1 / 3
  sorryAll goals completed! 🐙