Regular Covers of Open Relatively Compact Subanalytic Sets

Let U be an open relatively compact subanalytic subset of a real analytic manifold. We show that there exists a finite linear covering (in the sense of Guillermou and Schapira) of U by subanalytic open subsets of U homeomorphic to a unit ball. We also show that the algebra of open relatively compact subanalytic subsets of a real analytic manifold is generated by subsets subanalytically and bi-lipschitz homeomorphic to a unit ball. Let M be a real analytic manifold of dimension n. In this paper we study the algebra S (M) of relatively compact open subanalytic subsets of M . As we show this algebra is generated by sets with Lipschitz regular boundaries. More precisely, we call a relatively compact open subanalytic subset U ⊂ M an open subanalytic Lipschitz ball if its closure is subanalytically bi-Lipschitz homeomorphic to the unit ball of R. Here we assume that M is equipped with a Riemannian metric. Any two such metrics are equivalent on relatively compact sets and hence the above definition is independent of the choice of a metric. Theorem 0.1. The algebra S (M) is generated by open subanalytic Lipschitz balls. That is to say if U is a relatively compact open subanalytic subset of M then the characteristic function 1U is a linear combination of functions of the form 1W1 , ..., 1Wm , where the Wj are open subanalytic Lipschitz balls. Note that, in general, U cannot be covered by subanalytic Lipschitz balls, as it is easy to see for {(x, y) ∈ R; y < x, x < 1}, M = R, due to the presence of cusps. Nevertheless we show the existence of a ”regular” cover in the sense that we control the distance to the boundary. Theorem 0.2. Let U ∈ S (M). Then there exist a finite cover U = ⋃ i Ui by open subanalytic sets such that : (1) every Ui is subanalytically homeomorphic to an open n-dimensional ball; (2) there is C > 0 such that for every x ∈ U , dist(x,M \ U) ≤ C maxi dist(x,M \ Ui) The proofs of Theorems 0.1 and 0.2 are based on the regular projection theorem, cf. [6], [7], [8], the classical cylindrical decomposition, and the L-regular decomposition of subanalytic sets, cf. [4], [8], [9]. L-regular sets are natural multidimensional generalization of classical cusps. We recall them briefly in Subsection 1.6. We show also the following strengthening of Theorem 0.2. Theorem 0.3. In Theorem 0.2 we may require additionally that all Ui are open L-regular cells (i.e. interiors of L-regular sets). For an open U ⊂M we denote ∂U = U \ U . 1