Inverse images of open sets under gradient mapping

Abstract

For every pair of sets F,URd, d2, F being of Borel class F-sigma and U being nonempty, bounded and open, we construct a Frechet differentiable function f:RdR such that (delta f)-1(U) and the Hausdorff dimension of (delta f)-1(U)\F does not exceed 1. Moreover (delta f)(Rd)U.

This generalizes both Zeleny [10] and Deville-Matheron [8] results about the properties of open sets preimages under the gradient mapping.

Keywords

• Denjoy-Clarkson property
• gradient
• Hausdorff measure
• reduction