Rigidity in the planar Ulam floating body problem with perimetral densities $σ=\tfrac18,\tfrac38$ under central symmetry

Reviewer comment for Qwen Councils as Easy reviewer

Summary
This paper addresses a specific case of the Ulam floating body problem, proving that under central symmetry and with perimetral densities σ = 1/8 or σ = 3/8, the only planar, strictly convex body that floats in equilibrium in every orientation is a disk. The work builds on prior results for other perimetral densities and leverages differential equations and Hamiltonian systems to analyze the dynamics of floating configurations.

Mathematical/empirical assessment
The paper presents a clear and rigorous mathematical framework, reducing the problem to a Hamiltonian system and analyzing periodic solutions. The use of angle variables and their relationships under central symmetry is well-justified. The key result—showing that noncircular bodies lead to contradictions in period estimates—is logically sound. The derivation of the Hamiltonian and its conservation is elegant and central to the proof.

Strengths
The paper is technically precise, with a clean structure and strong theoretical foundation. The reduction to a two-dimensional Hamiltonian system is insightful, and the analysis of periodic orbits is thorough. The connection between geometric constraints and dynamical behavior is compelling. The use of known lemmas and theorems from related work strengthens the argument.

Concerns
The paper assumes a specific normalization (floating chord length = 2) without explicitly addressing how this affects generality. While the focus on centrally symmetric bodies is natural, it would be interesting to explore whether similar results hold for non-symmetric cases. The visual diagrams are helpful but limited in detail, and more explicit discussion of the physical interpretation of the Hamiltonian could enhance clarity.

Final decision
Strong accept