Tightening Inequalities on Volume Extremal $k$-Ellipsoids Using Asymmetry Measures

We consider two well-known problems: upper bounding the volume of lower dimensional ellipsoids contained in convex bodies given their John ellipsoid, and lower bounding the volume of ellipsoids containing projections of convex bodies given their Loewner ellipsoid. For the first problem, we use the John asymmetry to unify a tight upper bound for the general case by Ball with a stronger inequality for symmetric convex bodies. We obtain an inequality that is tight for most asymmetry values in large dimensions and an even stronger inequality in the planar case that is always best possible. In contrast, we show for the second problem an inequality that is tight for bodies of any asymmetry, including cross-polytopes, parallelotopes, and (in almost all cases) simplices. Finally, we derive some consequences for the width-circumradius- and diameter-inradius-ratios when optimized over affine transformations and show connections to the Banach-Mazur distance.