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In this paper we prove that the homothety conjecture holds true in the class of the convex bodies $B^n_p$, $1\\leq p\\leq \\infty$, the unit balls of $l_p^n$; namely, we show that $(B^n_p)_{\\d} = c B^n_p$ 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. 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Werner","submitted_at":"2009-11-03T17:38:58Z","abstract_excerpt":"Let $K$ be a convex body in $\\bbR^n$ and $\\d>0$. The homothety conjecture asks: Does $K_{\\d}=c K$ imply that $K$ is an ellipsoid? Here $K_{\\d}$ is the (convex) floating body and $c$ is a constant depending on $\\d$ only. In this paper we prove that the homothety conjecture holds true in the class of the convex bodies $B^n_p$, $1\\leq p\\leq \\infty$, the unit balls of $l_p^n$; namely, we show that $(B^n_p)_{\\d} = c B^n_p$ 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. 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