From the Mahler conjecture to Gauss linking integrals
Greg Kuperberg
TL;DR
The paper studies the Mahler conjecture for the product $v(K) = (\Vol K)(\Vol K^\circ)$ of a symmetric convex body $K \subset \mathbb{R}^n$ and its polar, by establishing a version of the bottleneck conjecture. The approach constructs a domain $K^\diamond \subseteq K \times K^\circ$ (built from necks $K^+$ and $K^-$ of the indefinite inner product space pseudospheres $H^+$ and $H^-$) and uses a Gauss-type linking form on a compactified space to derive a constant lower bound on $\Vol K^\diamond$, with equality when $K$ is an ellipsoid. The bottleneck conjecture implies the Mahler conjecture up to a factor $(\pi/4)^n \gamma_n$, where $\gamma_n$ is a monotone factor starting at $4/\pi$ and converging to $\sqrt{2}$, strengthening Bourgain–Milman’s result. The analysis reduces to a differential equation $f'' + (a+b)\tanh(\alpha) f' + ab f = 0$ with $f(0)=1$, $f'(0)=0$, and shows $f(\alpha) < 1$ for $\alpha \neq 0$, with equality only when the necks are orthogonal; the framework extends to joins of necks in the pseudospheres and connects to traditional Gauss linking integrals on $S^{n-1}$ and $H^{n-1}$.
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 $K^\diamond \subseteq K \times K^\circ$ 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 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 $c^n$. 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. 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 $S^{n-1}$ and in hyperbolic space $H^{n-1}$.