The bottleneck conjecture

The Mahler volume of a centrally symmetric convex body $K$ is defined as $M (K) = (Vol K) (Vol K^{\circ})$ . Mahler conjectured that this volume is minimized when $K$ is a cube. We introduce the bottleneck conjecture, which stipulates that a certain convex body $K^{♢} \subset K \times K^{\circ}$ 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

Mathematical Subject Classification

Primary: 52A40

Secondary: 46B20, 53C99

References

Forward citations

Publication

Received: 23 November 1998

Accepted: 20 May 1999

Published: 29 May 1999

Proposed: Robion Kirby

Seconded: Walter Neumann, Yasha Eliashberg

Authors

Greg Kuperberg
Department of Mathematics University of California at Davis One Shields Avenue Davis California 95616 USA

Volume 3, issue 1 (1999)

Greg Kuperberg