Piecewise Linearization of Real-valued Subanalytic Functions

We show that for a subanalytic function / on a locally compact subanalytic set X there exists a unique subanalytic triangulation (a simplicial complex K , a subanalytic homeomorphism n: \K\ —► X) such that f o n\a , a 6 K , are linear. Let X be a subanalytic set contained and closed in a Euclidean space. A subanalytic triangulation of X is a pair (K , n) where K isa simplicial complex and n is a subanalytic homeomorphism from the underlying polyhedron \K\ to X. Here we give \K\ a subanalytic structure by realizing K in a Euclidean space so that \K\ is closed in the Euclidean space. The existence of a subanalytic triangulation of X was shown by [2 and 4]. [2 and 4] use induction on the dimension of the ambient space like [5]. Moreover [10] proved the uniqueness up to PL homeomorphism, namely, that if there are two subanalytic triangulations (K ,it) and (K1 ,n) of X then \K\ and \K'\ are PL homeomorphic. Our purpose is the following theorem of a unique subanalytic triangulation of a subanalytic function. Theorem 1. Let f be a subanalytic function on X. Then there exists a unique subanalytic triangulation (K ,n) of X such that for every simplex o in K, f o n\a is linear. Here the uniqueness means that for another subanalytic triangulation (K1, ri) of X with the same property as (K , it) there exists a PL homeomorphism r from \K\ to \K'\ such that fon'ox = fon. I proved this in weaker forms in [9]. I constructed n using the integrations of vector fields. Hence triangulations of subanalytic functions in [9] were of class C only, and the uniqueness did not follow because of the failure of the topological Hauptvermutung [6]. As for this paper, we apply the projection method of [5] to the local proof of Theorem 1. The uniqueness theorem [10] pastes the local subanalytic triangulations and proves globally Theorem 1, which is similar to the proof of the theorem of C°° triangulation of C°° manifold (CairnsWhitehead, e.g. [7]). We use also the Alexander trick (see for its definition [8, 10] and the statement before Lemma 9). Teissier [11] also used the projection Received by the editors September 16, 1987. 1980 Mathematics Subject Classification (1985 Revision). Primary 32B25; Secondary 32B20.