Convex Floating Bodies of Equilibrium

TL;DR

This paper addresses Ulam's long-standing problem by proving that the Euclidean ball is the unique origin-symmetric convex body of uniform density that floats in equilibrium in any orientation in three dimensions.

Contribution

It extends the uniqueness result of floating bodies to higher dimensions within the class of origin-symmetric convex bodies with density 1/2.

Findings

Euclidean ball is unique in floating equilibrium in 3D

Results apply to origin-symmetric convex bodies in higher dimensions

Confirms Falconer's result for 3D case

Abstract

We study a long standing open problem by Ulam, which is whether the Euclidean ball is the unique body of uniform density which will float in equilibrium in any direction. We answer this problem in the class of origin symmetric n-dimensional convex bodies whose relative density to water is 1/2. For n=3, this result is due to Falconer.