It is shown that if C is an n-dimensional convex body then there is an affine image C ˜ of C for which |∂ C ˜ |/| C ˜ (n−1)/n is no larger than the corresponding expression for a regular n-dimensional ‘tetrahedron’. It is also shown that among n-dimensional subspaces of Lp (for eachp∈[1,∞]), l p n has maximal volume ratio.
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Keith Ball (1991) studied this question.