VOLUME INEQUALITIES FOR SUBSPACES OF Lp

A direct approach is used to establish both Ball and Barthe’s reverse isoperimetric inequalities for the unit balls of subspaces of Lp. This approach has the advantage that it completely settles all the open uniqueness questions for these inequalities. Affine isoperimetric inequalities generally have ellipsoids as extremals. The so called reverse affine isoperimetric inequalities usually have simplices — or in the symmetric case cubes and their polars — as their extremals. Symmetrization techniques, developed and promoted by Steiner well over a century ago, have been used to establish a variety of powerful affine isoperimetric inequalities. The reverse inequalities would turn out to be much harder to establish. They appeared to require some sort of anti symmetrization technique. By 1990, only one significant reverse inequality had been established in dimensions greater than two: the Rogers-Shephard difference-body inequality (see e.g., Schneider [42]). Unfortunately, the techniques employed by Rogers and Shephard could not be adapted to establish any of the other conjectured reverse inequalities. A breakthrough occurred in 1990 when Keith Ball connected John’s theorem characterizing the largest ellipsoid contained in a convex body (the John ellipsoid) with the Brascamp-Lieb inequality. The Brascamp-Lieb inequality had been developed to solve the best-constant problem for Young’s convolution inequality (see the excellent recent survey of Gardner [11]). Ball discovered a gorgeous reformulation of the Brascamp-Lieb inequality that seemed tailor-made to exploit the John ellipsoid. Ball’s normalized Brascamp-Lieb inequality has had a profound impact on convex geometric analysis (see, e.g., Ball [1, 2, 3], Bastero and Romance [5], Giannopoulos and Papadimitrakis [13], Giannopoulos, Milman, and Rudelson [12], Giannopoulos, Perissinaki, and Tsolomitis [14], Schechtman and Schmuckenschläger [40], Schmuckenschläger [41]). Mathematics Subject Classification. 52A40. Research supported, in part, by NSF Grant DMS–0104363.