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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)

Citations:

Zbl 0668.00010