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Lipschitz stratification of subanalytic sets. (English) Zbl 0819.32007

After the pioneering work of T. Mostowski [Diss. Math. 243, 46 pp. (1985; Zbl 0578.32020)], who showed the existence of Lipschitz stratifications in the complex analytic case, the author shows the existence of Lipschitz stratifications in the real, for subanalytic sets. This is by no means easy. The techniques of Mostowski do not apply in the real, although some ideas do.
The decomposition done by the author is similar to triangulation but technically more complicated. Some of the techniques existed already in [S. Łojasiewicz, “Ensembles semi-analytiques, multigraphic”, I.H.E.S., Bures-sur-Yvette (1965)], like \(L\)-regular sets and regular projections, but they are modernized and regular projections are chosen in a very subtle way (following T. Mostowski [loc. cit.]). Flattening techniques are delicate and everything is nicely put together.
The reviewer does not like the author’s treating of the bibliography. It is not recalled in any consistent way (references are often lacking), for instance [the reviewer, S. Łojasiewicz and J. Stasica, Bull. Acad. Pol. Sci., Ser. Sci. Math. 27, 529-536 (1979; Zbl 0435.32006)] should be quoted for the fiber cutting lemma, as it is done in [E. Bierstone and P. D. Milman, Publ. Math., Inst. Hautes Etud. Sci. 67, 5-42 (1988; Zbl 0674.32002)].
Except for this, the paper is really well written and very clear, despite the technicallity of the subject.

MSC:

32B20 Semi-analytic sets, subanalytic sets, and generalizations
32B25 Triangulation and topological properties of semi-analytic andsubanalytic sets, and related questions
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References:

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