#### Geometry & Topology, Vol. 3 (1999)
Paper no. 5, pages 119--135.

## The bottleneck conjecture

### Greg Kuperberg

**Abstract**.
The Mahler volume of a centrally symmetric convex body K is defined as
M(K)= (Vol K)(Vol K^dual). Mahler conjectured that this volume is
minimized when K is a cube. We introduce the bottleneck conjecture,
which stipulates that a certain convex body $K^diamond subset K X
K^dual has least volume when K is an ellipsoid. If true, the
bottleneck conjecture would strengthen the best current lower bound on
the Mahler volume due to Bourgain and Milman. We also generalize the
bottleneck conjecture in the context of indefinite orthogonal geometry
and prove some special cases of the generalization.
**Keywords**.
Metric geometry, euclidean geometry, Mahler conjecture, bottleneck conjecture, central symmetry

**AMS subject classification**.
Primary: 52A40.
Secondary: 46B20, 53C99.

**DOI:** 10.2140/gt.1999.3.119

**E-print:** `arXiv:math.MG/9811119`

Submitted to GT on 23 November 1998.
Paper accepted 20 May 1999.
Paper published 29 May 1999.

Notes on file formats
Greg Kuperberg

Department of Mathematics, University of California

One Shields Avenue, Davis, CA 95616, USA

Email: greg@math.ucdavis.edu

GT home page

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