On bodies floating in equilibrium in every orientation

TL;DR

This paper investigates Ulam's problem about whether a solid of uniform density that floats in water in all orientations must be a sphere, providing new results related to this longstanding question.

Contribution

It presents several new results concerning the conditions under which a solid of uniform density can float in equilibrium in all orientations.

Findings

Identifies conditions under which solids must be spherical to float in all orientations

Provides partial answers and new insights into Ulam's problem

Advances understanding of equilibrium shapes of floating bodies

Abstract

Ulam's problem 19 from the Scottish Book asks: {\it is a solid of uniform density which floats in water in every position necessarily a sphere?} We obtain several results related to this problem.