On subanalytic geometry

TL;DR

This survey explores the geometric properties of globally subanalytic sets, covering foundational theorems, tools, Lipschitz geometry developments, and geometric integration theory, highlighting their structure, invariants, and measure-theoretic aspects.

Contribution

It provides a comprehensive overview of subanalytic geometry, including new results on metric triangulations, bi-Lipschitz triviality, and measure-theoretic properties of subanalytic sets.

Findings

Existence of metric triangulations for subanalytic sets

Invariance of the link under bi-Lipschitz mappings

Development of Lipschitz conic structure and geometric integration results

Abstract

These notes constitute a survey on the geometric properties of globally subanalytic sets. We start with their definition and some fundamental results such as Gabrielov's Complement Theorem or existence of cell decompositions. We then give the main basic tools of subanalytic geometry, such as Curve Selection Lemma, Lojasiewicz's inequalities, existence of tubular neighborhood, Tamm's theorem (definability of regular points), or existence of regular stratifications (Whitney or Verdier). We then present the developments of Lipschitz geometry obtained by various authors during the four last decades, giving a proof of existence of metric triangulations, introduced by the author of these notes, definable bi-Lipschitz triviality, Lipschitz conic structure, as well as invariance of the link under definable bi-Lipschitz mappings. The last chapter is devoted to geometric integration theory, studying the Hausdorff measure of globally subanalytic sets, integrals of subanalytic functions, as well as the density of subanalytic sets (the Lelong number) and Stokes' formula.