The bottleneck conjecture (Q1296208)

Let \(V\) be an \(n\)-dimensional vector space and let \(V^\ast\) be the dual vector space. The inner product between V and \(V^\ast\) is denoted by \(\langle \cdot,\cdot\rangle.\) If \(K \subset V\) is a centrally symmetric convex body centered at the origin, then there is a convex body \(K^\circ =\{\vec y\in V^\ast|\langle K,\vec y\rangle\subseteq [-1,1]\}\) called the dual or polar body of \(K.\) The Mahler volume of \(K\) is defined as \(M(K) =\text{Vol }K\times K^\circ = (\text{Vol }K)(\text{Vol }K^\circ).\) The Mahler volume arises in the geometry of numbers and in functional analysis. Mahler conjectured that this volume is minimized when \(K\) is a cube. The author presents the bottleneck conjecture, which stipulates that a certain convex body \(K^\lozenge\subset K\times K^\circ\) 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. The bottleneck conjecture is also generalized in the context of indefinite orthogonal geometry. Some special cases of the generalization are proved.