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^o. 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 long ago 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 new proof of the Bourgain-Milman theorem that yields an exponential factor of (pi/4)^n. The proof minimizes a different volume at the opposite end of the space of convex bodies, i.e., at ellipsoids. The minimization argument is based on indefinite inner products and Gauss-type integrals for linking numbers.