THE BIRKHOFF-PORITSKY CONJECTURE FOR CENTRALLY-SYMMETRIC BILLIARD TABLES

The paper’s Theorem 1.1 states exactly the claim used by the model: if T has a rotational invariant curve α of rotation number 1/4 consisting of 4-periodic orbits and the annulus A between α and {δ=0} is foliated by continuous rotational invariant curves, then the boundary γ is an ellipse . The proof uses a non-standard generating function S in oriented-line coordinates (p,ϕ) with twist S12=(ρ sin δ)/2>0, where ρ=h''+h, alongside the classical chord-length generating function, matching the model’s step (i) . From the foliation, the authors obtain a measurable monotone invariant line subbundle on A (equivalently, no conjugate points) and show that its existence already forces γ to be an ellipse (Lemma 3.1 and Theorem 3.2), exactly the model’s step (ii) . The special geometry of the 4-periodic invariant curve is encoded in Theorem 4.1 (parallelogram/rectangle properties and identities linking d and h), aligning with step (iii) . Finally, a Hopf-type integral inequality over A yields inequality (14), which reduces to a sharp Wirtinger inequality for μ(ψ)=cos(2d(ψ)); equality forces μ to have only 0 and 2 harmonics and implies h^2(ψ)=R^2/2(1−A cos 2(ψ−ψ0)), the support function squared of an ellipse (step (iv)) . The model’s outline is thus correct and substantially coincides with the paper’s argument, with minor notational imprecision (using p for the support function) and a slight misattribution of the Fourier-mode restriction to the support function instead of μ or h^2.