Volume ratios and a reverse isoperimetric inequality

Abstract

It is shown that if $C$ is an $n$-dimensional convex body then there is an affine image $\widetilde C$ of $C$ for which $${|\partial \widetilde C|\over |\widetilde C|^{n-1\over n}}$$ is no larger than the corresponding expression for a regular $n$-dimensional ``tetrahedron''. It is also shown that among $n$-dimensional subspaces of $L_p$ (for each $p\in [1,\infty]), \ell^n_p$ has maximal volume ratio.\vskip3in