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A closely related problem, considering the $(n-1)$-dimensional surface area instead of the diameter of the body, has been solved by Keith Ball, Volume ratios and a reverse isoperimetric inequality. J. London Math. Soc. (2) 44 (1991), no. 2, 351–359 [MR1136445]. The extreme case, as expected, is the $n$-dimensional simplex. Ball's proof may also work for the diameter. The article is available on the arXiv, see http://arxiv.org/abs/math/9201205 . The 2-dimensional case was solved long before, with a very short, elementary proof.