Tame geometry with application in smooth analysis (Q1424147)

This monograph is devoted to several related topics: geometry of real semialgebraic and tame sets, behavior of their ``metric'' characteristics under polynomial mappings, integral geometry, and geometry of critical and near critical values of differentiable mappings. Its subsequent chapters deal with the entropy dimension of a set, multidimensional variations and their use for bounding the \(\varepsilon\)-entropy of a set, the number of connected components of plane sections of semialgebraic sets, multidimensional variations of semialgebraic and tame sets, the multidimensional variations of images under polynomial mappings and a quantitative Sard theorem, quantitative transversality and cuspical values, quantitative Morse-Sard theorem bounding the \(\varepsilon\)-entropy of near-critical values, and finally some applications. It is an interesting and useful contribution to the study of the quantitative aspects of the Morse-Sard theorem.