On the volume of the John-L\"owner ellipsoid
Abstract
We find an optimal upper bound on the volume of the John ellipsoid of a k-dimensional section of the n-dimensional cube, and an optimal lower bound on the volume of the L\"owner ellipsoid of a projection of the n-dimensional cross-polytope onto a k-dimensional subspace. We use these results to give a new proof of Ball's upper bound on the volume of a k-dimensional section of the hypercube, and of Barthe's lower bound on the volume of a projection of the n-dimensional cross-polytope onto a k-dimensional subspace. We settle equality cases in these inequalities. Also, we describe all possible vectors in n, whose coordinates are the squared lengths of a projection of the standard basis in n onto a k-dimensional subspace.
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