Geometric and Functional Analysis

, Volume 18, Issue 3, pp 870–892

From the Mahler Conjecture to Gauss Linking Integrals

Article

DOI: 10.1007/s00039-008-0669-4

Cite this article as:
Kuperberg, G. GAFA Geom. funct. anal. (2008) 18: 870. doi:10.1007/s00039-008-0669-4

Abstract.

We establish a version of the bottleneck conjecture, which in turn implies a partial solution to the Mahler conjecture on the product v(K)  = (Vol K)(Vol K°) of the volume of a symmetric convex body \(K \in {\mathbb{R}}^{n}\) and its polar body K°. The Mahler conjecture asserts that the Mahler volume v(K) is minimized (non-uniquely) when K is an n-cube. The bottleneck conjecture (in its least general form) asserts that the volume of a certain domain \(K^{\diamond} \subseteq K \times K^{\circ}\) is minimized when K is an ellipsoid. It implies the Mahler conjecture up to a factor of (π/4)nγn, where γn is a monotonic factor that begins at 4/π and converges to \({\sqrt2}\). This strengthens a result of Bourgain and Milman, who showed that there is a constant c such that the Mahler conjecture is true up to a factor of cn.

The proof uses a version of the Gauss linking integral to obtain a constant lower bound on Vol K, with equality when K is an ellipsoid. It applies to a more general conjecture concerning the join of any two necks of the pseudospheres of an indefinite inner product space. Because the calculations are similar, we will also analyze traditional Gauss linking integrals in the sphere Sn-1 and in hyperbolic space Hn-1.

Keywords and phrases:

convexMahler conjectureGauss linking integralBourgain-Milman theorem

AMS Mathematics Subject Classification:

52A53 (46B07, 53A05)

Copyright information

© Birkhaeuser 2008

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of CaliforniaDavisUSA