Paata Ivanisvili @PI010101
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Paata Ivanisvili @PI010101 · 13h The preimage of every line under a conformal map of the unit disk has total length at most π², and this is best possible arxiv.org/pdf/2608.12844 I first learned about this problem from John Garnett and Donald Marshall's wonderful book Harmonic Measure. Chapter I gives the previously known suboptimal bound 4π. A later result showed that the optimal constant is strictly smaller than 4π, and that remained the state of the art until today. AI did the job. My contribution was to direct it toward the right problem, verify the argument, digest it, and present the solution in a short and hopefully easily readable form. The complete proof is now a little under four pages long. It is a really nice solution. My first reaction was: "Wow, how was this missed?" I remember having the same feeling when I first read the proof of the Sensitivity Conjecture. [embedded image of a textbook/paper excerpt] 5. The Hayman–Wu Theorem We give a very elementary proof, based on an idea of the late K. Øyma [1992], of the theorem of Hayman and Wu. The Hayman–Wu theorem will be a recurrent topic throughout this book. Theorem 5.1 (Hayman–Wu). Let φ be a conformal mapping from 𝔻 to a simply connected domain Ω and let L be any line. Then length(φ⁻¹(L ∩ Ω)) ≤ 4π. (5.1) Hayman and Wu [1981] gave the first proof of (5.1) with 4π replaced by some large unknown constant. Øyma [1992] obtained the constant 4π, Rohde [2002] proved that the best constant in (5.1) is strictly smaller than 4π, and Øyma [1993] proved that the best constant is at least π². The sharp constant in (5.1) is not known. See Exercises 24 and VI.3. We present Øyma's elementary proof, as modified by Rohde.
Note from Claude Sonnet 5
Tweet by Paata Ivanisvili claiming an AI solved the sharp constant (π²) for the Hayman-Wu theorem, with a screenshot of a textbook excerpt (Garnett & Marshall, Harmonic Measure) stating the theorem and its proof history embedded below the text.