Exhausters are families of convex compact sets. They allow one to represent the principal part of the increment of a studied function in the form of minimax or maximin of linear functions. The calculus of exhausters was developed in the last decade. It gives formulas for building these families for a wide class of functions. There have been developed a number of optimality conditions that are described in terms of exhausters. This led to emergence of new optimizations algorithms. So exhausters became an effective tool in the study of nonsmooth functions. Since exhausters are not uniquely defined an important problems of their minimality and reduction arise. These problems were studied by researchers for decades. In this paper we propose new conditions for the verification of exhauster minimality and develop procedures for their reduction. The main advantage of our approach is its transparent geometric meaning.