In this paper, a new approach to robustness problems is described which addresses large classes of systems with performance specifications depending nonlinearity on a vector q of uncertain parameters in a hypercube Q. Whereas verification of a system's robust requirement f(q) < 0 for all q /spl isin/ Q may be prohibitive, various "dilation integrals" /spl Phi//sub k/ involving f(q) over Q are often straightforward to compute in closed form and provide a sharp upper bound on the volume of violation in the parameter set Q. Since computational difficulty generally increases with k, one of the focal points of this paper are the results related to the size of the required k in order to achieve some small prespecified volume of violation /spl epsi/ > 0. Such results are described in terms of an underlying conditioner 0 /spl les/ /spl theta/ /spl les/ 1 and the dimension n of q. For large classes of problems, as n grows, the required k value may be quite low.