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.

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

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