Density of semi-pfaffian sets
type: Item
instance of: scholarly article
title: Density of semi-pfaffian sets
zbMATH Open document ID: 0917.32003
author: Jean-Marie Lion
published in: Annales de la Faculté des Sciences de Toulouse. Mathématiques. Série VI
publication date: 1999-08-02T00:00:00Z
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/73447
review text: 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.
Mathematics Subject Classification ID: 32C05
Mathematics Subject Classification ID: 32C25
Mathematics Subject Classification ID: 32B20
zbMATH DE Number: 1236377
zbMATH Keywords: semi-pfaffian sets
zbMATH Keywords: density
zbMATH Keywords: integration
reviewed by: Zofia Denkowska
MaRDI profile type: Publication
OpenAlex ID: W2327020019
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DOI: 10.5802/AFST.890
description: scientific article; zbMATH DE number 1236377