Nonnegative Whitney Extension Problem for $C^1({\mathbb{R}}^n)$

Abstract

Let $f$ be a real-valued function on a compact subset in ${\mathbb{R}}^n$. We show how to decide if $f$ extends to a nonnegative and $C^1$ function on ${\mathbb{R}}^n$. There has been no known result for nonnegative $C^m$ extension from a general compact set $E$ when $m>0$. The nonnegative extension problem for $m\ge 2$ remains open.

arXiv comment10 pages