Liam Price

math.MG — Metric Geometry

Analytic symmetric floating bodies

Contributed by Liam Price, Tom de Groot

In every dimension $d\ge3$, 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 $(d+4)^{-1/2}$, 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 $\pi$. 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.