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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
##### Keywords:
density; preparation theorem; subanalytic sets