From the Mahler conjecture to Gauss linking integrals

The Mahler volume of a centrally symmetric convex body $K$ in $n$ dimensions is defined as the product of the volume of $K$ and the polar body $K^\circ$. It is an affinely invariant number associated to a centrally symmetric convex body, or equivalently a basis-independent number associated to a finite-dimensional Banach space. Mahler conjectured that the Mahler volume in $n$ dimensions is maximized by ellipsoids and minimized by cubes. The upper bound was proven by Santalo. Bourgain and Milman showed that the lower bound, known as the Mahler conjecture, is true up to an exponential factor. Their theorem is closely related to other recent results in high-dimensional convex geometry. I will describe a proof of the Bourgain-Milman theorem that yields an exponential factor of $(\pi/4)^n$. The idea is to minimize a different volume at the opposite end of the space of convex bodies, i.e., at ellipsoids. The proof of the minimum is a calibration argument that uses a Gauss linking integral.