Given a set $S$ which is the zeroes of a finite number of homogenous polynomials in $x\in\mathbb{R}^n$, I want a constant $\alpha$ and a $C^2$ approximation, denoted $d$, to the function $d(x,S)=\inf_{y\in S}\parallel x-y\parallel_2$ with bounded first and second derivatives such that $\sup_{x\in\mathbb{R}^n}|d(x)-d(x,S)|\leq \alpha$.

My initial approach was to first stratify $S$ into a finite number of Nash submanifolds $\{S_j\}_j$, each of which has a tubular neighbourhood $\{U_j\}_j$ on which $d(x,S_j)$ is analytic. I then tried to mollify between these, which was hard in $\mathbb{R}^n$, and I also tried a smooth approximation to the minimum of these functions which also didn't work. And Stone-Weierstrass isn't useful as I need these bounds on all of $\mathbb{R}^n$.

  • $\begingroup$ This seems like it's really a real analysis question - the distance function to a closed set is actually Lipschitz continuous, and you're asking whether a Lipschitz continuous function can be approximated in the sup-norm by a $C^2$ function. I strongly suspect the answer is yes, but I do not have a reference offhand. Perhaps removing the submanifold tag and retagging as real-analysis would provide exposure to people with more specific knowledge. $\endgroup$ – KReiser Mar 26 '19 at 21:13

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