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Analytic symmetric floating bodies

A mix of human-written and AI-generated textHuman understanding: some partsmath.MG — Metric Geometrymath.CA — Classical Analysis and ODEs

Contributed by Liam Price ↗, Tom de Groot ↗

Version 1 / Oct 29, 2026 / CC BY 4.0

Abstract

In every dimension d≥3d\ge3, we construct nonspherical, origin-symmetric, strictly convex bodies of revolution with real-analytic boundary that float in equilibrium in every orientation. Their densities tend to the unit-ball cap fraction at height (d+4)−1/2(d+4)^{-1/2}, where we prove that a degree-four spherical resonance is isolated. More generally, our construction applies at isolated even-degree resonances whose waterline angle is an irrational multiple of π\pi. Varying the density gives formal deformations. To obtain actual bodies, we use a finite-order diagonal reduction to establish inverse estimates away from small divisors. Newton iteration then corrects fixed-order truncations on closed parameter sets of asymptotically full measure.

Provenance statement

The original argument for the four-dimensional case was developed by GPT-6 Pro in ChatGPT. GPT-6.1 Sol and GPT-6 Astra, working in Codex, subsequently extended the argument to all dimensions d ≥ 3. The Codex models also assisted with drafting, revising, and proofreading the manuscript.

Tools used

OpenAI
ChatGPTVersion GPT 6 Pro
OpenAI
CodexVersion GPT 6.1 Sol & GPT 6 Astra

References

  1. F. W. J. Olver, A. B. Olde Daalhuis, D. W. Lozier, et al., eds., NIST Digital Library of Mathematical Functions , https://dlmf.nist.gov/, Chapter 18.link
  2. D. I. Florentin, C. Schütt, E. M. Werner, and N. Zhang, Convex floating bodies of equilibrium , Proc. Amer. Math. Soc. 150 (2022), 3037--3048. doi:10.1090/proc/15697 (https://doi.org/10.1090/proc/15697).DOI
  3. E. M. Matveev, An explicit lower bound for a homogeneous rational linear form in the logarithms of algebraic numbers. II , Izv. Math. 64 (2000), 1217--1269. doi:10.1070/IM2000v064n06ABEH000314 (https://doi.org/10.1070/IM2000v064n06ABEH000314).DOI
  4. D. Ryabogin, A negative answer to Ulam's Problem 19 from the Scottish Book , Ann. of Math. (2) 195 (2022), 1111--1150. doi:10.4007/annals.2022.195.3.5 (https://doi.org/10.4007/annals.2022.195.3.5).DOI
  5. F. Wegner, Floating bodies of equilibrium in three dimensions. The central symmetric case , arXiv:0803.1043v2 (https://arxiv.org/abs/0803.1043v2), 2009.arXiv

Version history

  1. v1Submitted by Liam PriceInitial depositCurrentOct 28, 2026