Rigidity in the Planar Ulam Floating Body Problem with perimetral density $\sigma=\tfrac16$

TL;DR

This paper proves that, for convex domains with a specific perimetral density, the only shape floating in equilibrium in all positions is the disk, establishing a new rigidity result.

Contribution

It introduces a novel approach using Zindler carousels to analyze the Ulam floating body problem with rational perimetral densities, proving disk uniqueness.

Findings

The disk is the only convex domain floating in equilibrium for perimetral density 1/6.

Reduction of the problem to a dynamical system associated with an inscribed equilateral hexagon.

Provides a new rigidity result for rational perimetral densities in convex geometry.

Abstract

We study the two-dimensional Ulam's floating body problem for convex domains with perimetral density $\sigma=\tfrac16$. Using the framework of Zindler carousels, we reduce the problem to a two-dimensional dynamical system associated with an inscribed equilateral hexagon. Our main result shows that the disk is the only convex domain floating in equilibrium in every position for this perimetral density. This provides a new rigidity result for rational perimetral densities in the convex setting.