On the local geometry of definably stratified sets

2017/01/18 by Trotman, David, Valette, Guillaume
#03C10 #32S15 (Primary) #58A35 (Secondary) #Differential Geometry (math.DG) #FOS: Mathematics #Logic (math.LO) #Metric Geometry (math.MG)

paper · doi:10.48550/arxiv.1701.05087

Abstract

We prove that a theorem of Pawlucki, showing that Whitney regularity for a subanalytic set with a smooth singular locus of codimension one implies the set is a finite union of differentiable manifolds with boundary, applies to definable sets in polynomially bounded o-minimal structures. We give a refined version of Pawlucki's theorem for arbitrary o-minimal structures, replacing Whitney (b)-regularity by a quantified version, and we prove related results concerning normal cones and continuity of the density. We analyse two counterexamples to the extension of Pawlucki's theorem to definable sets in general o-minimal structures, and to several other statements valid for subanalytic sets. In particular we give the first example of a Whitney (b)-regular definably stratified set for which the density is not continuous along a stratum.

Related

• A geometric proof of the existence of definable Whitney stratifications
• Whitney stratifications and the continuity of local Lipschitz Killing curvatures
• Regular vectors and bi-Lipschitz trivial stratifications in o-minimal structures
• Tangent cones and C¹ regularity of definable sets
• A geometric proof of the existence of Whitney stratifications