On the homothety conjecture

Let K be a convex body in R_n and δ > 0. The homothety conjecture asks: Does Kδ = cK imply that K is an ellipsoid? Here Kδ is the (convex) floating body and c is a constant depending on δ only. In this paper we prove that the homothety conjecture holds true in the class of the convex bodies Bⁿ_p , 1 ≤ p ≤ ∞, the unit balls of ℓⁿ_p; namely, we show that (Bⁿ _p)δ = cBⁿ_p if and only if p = 2. We also show that the homothety conjecture is true for a general convex body K if δ is small enough. This improves earlier results by Schütt and Werner [16] and Stancu [20].