Density of semi-pfaffian sets
Semi-pfaffian sets introduced by R. Moussu in 1988, following and idea of Hovansky, has developed ever since, mostly in Dijon (Moussu, Rocke, Lion, Rolin), forming a very useful theory. In ordinary differential equations, even if the equation is analytic, the solutions are very rarely subanalytic but they are mostly semi-pfaffian [cf. \textit{R. Moussu} and \textit{C.-A. Roche}, Invent. Math. 105, No. 2, 431-441 (1991; Zbl 0769.58050)]. In the present paper the author proves the existence of densities in each boundary point of a semi-pfaffian set, generalizing therefore the results of Kurdyka-Poly-Raby [\textit{K. Kurdyka}, \textit{J.-B. Poly} and \textit{G. Raby}, Lect. Notes Math. 1420, 170-177 (1990; Zbl 0694.32001); \textit{K. Kurdyka} and \textit{G. Raby}, Inst. Fourier 39, No. 3, 753-771 (1989; Zbl 0673.32015)] about subanalytic sets, as well as these of Roche [\textit{C.-A. Roche}, Astérisque 222, 373-387 (1994; Zbl 0831.32004)]. The methods used are these of integral geometry (cf. the book [\textit{L. A. Santaló}, `Integral geometry and geometric probability' (1976; Zbl 0342.53049)] and the work of Langevin [\textit{R. Langevin}, `Un peu de géométrie intégrale, Images des Mathématiques, CNRS, 58-67 (1995)]), namely the Cauchy-Crofton integration formula, and adaption of the subanalytic proof. The paper is very clear, well written, and an exhaustive bibliography makes it easy to follow the author's thought.
scientific article; zbMATH DE number 1236377