Log-analytic nature of the volume of subanalytic sets
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.
scientific article; zbMATH DE number 1553608