Rigidity in the planar Ulam floating body problem with perimetral density

Source abstract

Abstract We study the 2D Ulam’s floating body problem for convex domains with perimetral density $\sigma =\tfrac 16$. Using the framework of Zindler carousels, we reduce the problem to a 2D 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.