Logarithmic reduction of the level of randomness in some probabilistic geometric constructions

Many of the surprising phenomena occurring in high dimensions are proved by use of probabilistic arguments, which show the existence of organized and regular structures but do not hint as to where exactly do these structures lie. It is an intriguing question whether some of them could be realized explicitly. In this paper we show that the amount of randomness used can be reduced significantly in many of these questions from asymptotic convex geometry, and most of the random steps can be substituted by completely explicit algorithmic steps. The main tool we use is random walks on expander graphs.