## Volumes of sections of cubes and related problems.(English)Zbl 0674.46008

Geometric aspects of functional analysis, Isr. Semin., GAFA, Isr. 1987-88, Lect. Notes Math. 1376, 251-260 (1989).
[For the entire collection see Zbl 0668.00010.]
The volume ratio of a symmetric convex body $$C\subset {\mathbb{R}}^ n$$ is the number $vr(C)=(vol\quad C/vol\quad E)^{1/n},$ where vol C $$=$$ volume of C and vol E $$=$$ volume of the ellipsoid E of maximum volume contained in C. A classical result of F. John shows that $$vr(c)<\sqrt{n}$$ for every such symmetric n-dimensional convex body C.
In this paper, it is shown that $$vr(C)\leq vr(Q_ n)=(2/V_ e^{1/2})$$, where $$Q_ n$$ is the n-dimensional cube in $${\mathbb{R}}^ n$$ and $$V_ n$$ is the volume of the unit ball in $${\mathbb{R}}^ n$$. A reformulation of this result in terms of volumes of sections of $$Q_ n$$, a derivation of the best possible upper bound for the volume of such a section, and an “isomorphic version” of the above inequality which involves the so- called cubical volume ratio of C are also given.
Reviewer: J.R.Holub

### MSC:

 46B20 Geometry and structure of normed linear spaces 52C07 Lattices and convex bodies in $$n$$ dimensions (aspects of discrete geometry)

Zbl 0668.00010