Mild parametrizations of power-subanalytic sets

Abstract

We obtain two uniform parametrization theorems for families of bounded sets definable in R R an . Let X = {X t ⊂ (0, 1) n | t ∈ T } be a definable family of sets X t of dimension at most m. Firstly, X t admits a C r -parametrization consisting of cr m maps for some positive constant c = c(X), which is uniform in t. Secondly, X t admits a C-mild parametrization for any C > 1, which is also uniform in t.