On the homothety conjecture for the body of flotation and the body of buoyancy on a plane

TL;DR

This paper explores homothety conjectures for convex bodies in a plane, establishing conditions under which bodies of flotation and buoyancy are homothetic, and deriving related geometric properties and affine analogues of classical theorems.

Contribution

It introduces new conditions linking homothety of flotation and buoyancy bodies to ellipses, and develops affine counterparts of classical floating body theorems using differential geometry.

Findings

If flotation and buoyancy bodies are homothetic with specific chord conditions, the body is an ellipse.

Derived fundamental properties of flotation, buoyancy, and illumination bodies.

Established affine versions of classical floating body theorems and Zindler carousels.

Abstract

We investigate several closely related "homothety conjectures" for convex bodies on a plane. Using the modern language of differential geometry, we systematically derive the fundamental properties of bodies of flotation, bodies of buoyancy, and bodies of illumination. As a direct consequence of our results, we show that if the body of flotation is homothetic to the body of buoyancy, and if every chord of flotation cuts off from the boundary exactly $\frac{1}{3}$ of its total affine arc length, then $K$ is an ellipse. We also provide natural affine counterparts of the classical theorems on the floating body problem from the Scottish Book due to H. Auerbach. In particular, we obtain an affine counterpart of Zindler carousels introduced by J. Bracho, L. Montejano, and D. Oliveros.