Mild parametrizations of power-subanalytic sets

Abstract

We obtain two uniform parametrization theorems for families of bounded sets definable in ${\mathbb{R}}_{an}^{\mathbb{R}}$. Let $X=\{X_t\subset (0,1{)}^n\mid t\in T\}$ be a definable family of sets $X_t$ of dimension at most $m$. Firstly, $X_t$ admits a $C^r$-parametrization consisting of $c{r}^m$ 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 $t$.

References in corpus (3)

• Asymptotic $h$-expansiveness rate of $C^{\infty }$ maps
• Smooth parameterizations of power-subanalytic sets and compositions of Gevrey functions
• On a family of mild functions