The Bourgain-Milman inequality is an (asymptotic) inverse to the classical Santaló inequality: the latter states that the volume product vol(K)vol(K^o) of symmetric convex bodies K (or more generally, convex bodies K with barycentre at the origin) is maximised by ellipsoids (here K^o is the polar body of K), while Bourgain and Milman proved in 1987 that vol(K)vol(K^o) ? c^n vol(D)^2 for every n-dimensional convex body K as above, where D here is the n-dimensional unit Euclidean ball and c is a constant independent of the dimension n.
Until recently, all proofs of this fact relied heavily on deep results from areas other than Convex Geometry, such as Operator Theory, Differential Geometry or Harmonic Analysis. We will present an alternative proof that uses only convex-geometric tools, and in particular methods developed for the purpose of studying isotropic convex bodies.
This is joint work with A. Giannopoulos and G. Paouris.