Let K be a convex body in Rn and Kd its floating body. The homothety conjecture asks: "Does Kd=c K imply that K is an ellipsoid?" Here c is a constant depending on d only. We prove that the homothety conjecture holds true in the class of the convex bodies Bnp, 1 = p = 8, the unit balls of lpn; namely, we show that (Bnp)d = c Bnp if and only if p=2. We also show that the homothety conjecture is true for a general convex body K if d is small enough.