Revista de la
Unión Matemática Argentina
Two results on the homothety conjecture for convex bodies of flotation on the plane
M. Angeles Alfonseca, Fedor Nazarov, Dmitry Ryabogin, Alina Stancu, and Vladyslav Yaskin

Volume 69, no. 1 (2026), pp. 93–108    

Published online (final version): December 9, 2025

https://doi.org/10.33044/revuma.5089

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Abstract

We investigate the homothety conjecture for convex bodies of flotation of planar domains close to the unit disk $B$. We show that for every density $\mathcal{D}\in (0,\frac{1}{2})$, there exists $\gamma=\gamma(\mathcal{D})>0$ such that if $(1-\gamma)B\subset K\subset (1+\gamma)B$ and the convex body of flotation $K^\mathcal{D}$ of an origin-symmetric body $K$ of density $\mathcal{D}$ is homothetic to $K$, then $K$ is an ellipse. On the other hand, we also show that if the symmetry assumption is dropped, then there is an infinite set of densities accumulating at $\frac{1}{2}$ for which there is a body $K$ different from an ellipse with the property that $K^{\mathcal{D}}$ is homothetic to $K$.

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