The purpose of functional tolerancing process is to define the geometrical specifications of parts ensuring functional requirements. For automotive and aircraft industries, the tolerance synthesis has become an important issue in product design process. Indeed, designers need methods and rules to determine the tolerances. To define rules, a mathematical formulation of tolerance synthesis is detailed. This mathematical formulation of tolerance synthesis simulates the influences of geometrical deviations on the geometrical behavior of the mechanism, and integrates the quantifier notion (existential quantifier ‘there exists’ and universal quantifier ‘for all’). It takes into account not only the influence of geometrical deviations but also the influence of the types of contacts on the geometrical behavior of the mechanism; these physical phenomena are modeled by convex hulls (compatibility hull, interface hull and functional hull) which are defined in parametric space. With this description by convex hulls, a mathematical expression of the admissible deviations of parts integrates the quantifier notion. This notion translates the concept that a functional requirement must be respected in at least one acceptable configuration of gaps (existential quantifier there exists), or that a functional requirement must be respected in all acceptable configurations of gaps (universal quantifier for all). With this approach, some rules are formalized to determine the modifier (maximum material condition or minimum material condition) function from the type of quantifier. These rules have been performed with success in French automotive industries.< Réduire