A Deterministic Proof of the Slicing Theorem

Abstract

We give a deterministic proof of Bourgain's slicing theorem. The proof uses neither stochastic localization nor stochastic calculus, does not rely on Guan's covariance bound, and avoids $M$-ellipsoids, Pisier's regularization machinery, and complex interpolation. Instead, the argument is based on a deterministic one-parameter quadratic perturbation of the measure, a covariance-survival principle, and a geometric analysis of $L_p$-centroid bodies.