John functions for o-minimal domains

Abstract

Abstract A John function is a continuously differentiable function whose gradient is bounded by the reciprocal of the Euclidean distance to the boundary of the domain. Here we construct John functions whose gradient norms have positive lower bounds and whose domain is a definable set of an o-minimal structure. We prove their definability and give explicit estimates for a number of John functions required for a given set to obtain uniform positive lower bounds.