Two results on the homothety conjecture for convex bodies of flotation on the plane

Joint with M. Angeles Alfonseca, Dmitry Ryabogin, Alina Stancu, and Vladyslav Yaskin.

We investigate the homothety conjecture for convex bodies of flotation of bodies close to the unit disk in the plane. We show that for every density D ∈ (0, 1/2 ), there exists γ = γ(D) > 0 such that if (1 − γ)B ⊂ K ⊂ (1 + γ)B and the convex body of flotation K_D of an origin symmetric body K of density 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 1/2 for which there is a body K different from an ellipse with the property that K_D is homothetic to K.

This is the last in the series of three lectures.