Real equisingularity: Lelong numbers and polar projections

We explain how real analogues of codimension 1 complex polar varieties, and their images under the projections defining them, can be used to obtain an equisingularity condition in subanalytic geometry which is a real version of equimultiplicity: the continuity of the Lelong numbers. It follows that these numbers are continuous along Verdier strata; extending to the reals a theorem of Hironaka. The proof is based on a Cauchy-Crofton formula for th e Lelong numbers of subanalytic sets. (C) 2000 Editions scientifiques et me dicales Elsevier SAS.