2017/07/14 by Ivanov, Grigory · 1 citation
#FOS: Mathematics #Functional Analysis (math.FA)
paper · doi:10.48550/arxiv.1707.04442
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 Löwner 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 \Rn, whose coordinates are the squared lengths of a projection of the standard basis in \Rn onto a k-dimensional subspace.