This is an announcement for the paper "From the Mahler conjecture to Gauss linking integrals" by Greg Kuperberg.
Abstract: We establish a version of the bottleneck conjecture, which in turn implies a partial solution to the Mahler conjecture on the product $v(K) = (\Vol K)(\Vol K^\circ)$ of the volume of a symmetric convex body $K \in \R^n$ and its polar body $K^\circ$. 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 $K^\diamond \subset K \times K^\dual$ is minimized when $K$ is an ellipsoid. It implies the Mahler conjecture up to a factor of $(\pi/4)^n \gamma_n$, where $\gamma_n$ is a monotonic factor that begins at $4/\pi$ and converges to $\sqrt{2}$. This strengthen 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 $c^n$. The proof uses a version of the Gauss linking integral to obtain a constant lower bound on $\Vol K^\diamond$, with equality when $K$ is an ellipsoid. The proof applies to a more general bottleneck conjecture concerning the join of any two necks of complementary pseudospheres in an indefinite inner product space. Because the calculations are similar, we will also analyze traditional Gauss linking integrals in the sphere $S^{n-1}$ and in hyperbolic space $H^{n-1}$.
Archive classification: Metric Geometry; Functional Analysis
Remarks: 9 pages, 4 figures
The source file(s), mahler.tex: 52417 bytes, is(are) stored in gzipped form as 0610904.gz with size 18kb. The corresponding postcript file has gzipped size 65kb.
Submitted from: greg@math.ucdavis.edu
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