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Log-analytic nature of the volume of subanalytic sets. (Nature log-analytique du volume des sous-analytiques.) (French) Zbl 0982.32009
A subset \(Y\) of \(\mathbb{R}^m\) is called subanalytic if there exist a number \(d\in \mathbb{N}\) and a seminanalytic subset \(Z\) of the torus \(\mathbb{P}^{m+d}\) such that \(Y =\pi(Z)\cap \mathbb{R}^m\) by which \(\pi\) denotes the natural projection. Let \(D\) be a subanalytic subset of \(\mathbb{R}^m\). A function \(f : D\to \mathbb{R}\) is called globally subanalytic if its graph is globally subanalytic.
The authors prove the following Theorem: Let \(Y\) be a globally subanalytic subset of \(\mathbb{R}^{n+m}\) with fibers \(Y_x = Y\cap (\{x\}\times \mathbb{R}^m)\) of dimension at most \(k\). Let \(B\) be the set of points \(x\in \mathbb{R}^n\) such that the \(k\)-dimensional volume \(v(x)\) of \(Y_x\) is finite. Then \(B\) is a globally subanalytic subset of \(\mathbb{R}^n\) and \(v|B\) is of the form \(v = P(A_1,\dots,A_r,\log A_1,\dots,\log A_r)\) by which \(P\) is a polynomial and the \(A_i\) are globally subanalytic functions.
From this result the authors deduce a corollary about the log-analytic nature of the \(k\)-dimensional density of globally subanalytic subsets of \(\mathbb{R}^m\) with dimension at most \(k\).
If in the theorem \(Y\) is semialgebraic then \(B\) is semialgebraic, too.
Important for the proof of the theorem are a preparation theorem for subanalytic functions and Lipschitz stratification for compact subanalytic sets.

MSC:
32B20 Semi-analytic sets, subanalytic sets, and generalizations
14P15 Real-analytic and semi-analytic sets
14P10 Semialgebraic sets and related spaces
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