On the volume of the John-L\"owner ellipsoid

Grigory Ivanov

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arXiv · 1707.04442

On the volume of the John-L\"owner ellipsoid

Grigory Ivanov

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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 $\R^n,$ whose coordinates are the squared lengths of a projection of the standard basis in $\R^n$ onto a $k$-dimensional subspace.

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Grigory Ivanov. 2017-07-14. On the volume of the John-L\"owner ellipsoid. https://arxiv.org/abs/1707.04442

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