On the Homothety Conjecture

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Let K be a convex body in ⁿ and >0. The homothety conjecture asks: Does K=c K imply that K is an ellipsoid? Here K is the (convex) floating body and c is a constant depending on only. In this paper we prove that the homothety conjecture holds true in the class of the convex bodies Bⁿₚ, 1≤ p≤ ∞, the unit balls of lₚⁿ; namely, we show that (Bⁿₚ) = c Bⁿₚ if and only if p=2. We also show that the homothety conjecture is true for a general convex body K if is small enough. This improvs earlier results by Schütt and Werner SW1994 and Stancu Stancu2009.