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

TL;DR

This paper establishes optimal bounds on the volumes of John and L"owner ellipsoids related to sections and projections of high-dimensional cubes and cross-polytopes, providing new proofs and characterizing equality cases.

Contribution

It introduces new optimal bounds for ellipsoid volumes of sections and projections, and characterizes all vectors corresponding to squared lengths of projected basis vectors.

Findings

Optimal upper bound on John ellipsoid volume of a cube section

Optimal lower bound on L"owner ellipsoid volume of a cross-polytope projection

Complete characterization of vectors of squared projection lengths

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.