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Consider a set $\Omega\subseteq\mathbb{R}^n$ being "sufficiently regular", for example being the image of a $C^1$ mapping from $\mathbb{R}^p$ for some $p\ge1$. We may then consider the mapping $$ g:\mathbb{R}^n \to \mathbb{R}^n \\ g(y) = \mathrm{argmin}_{\theta\in\Omega}\|y - \theta\|_2, $$ where $\mathrm{argmin}$ denotes the argument minimum. It may be that the argument minimum is not unique, hopefully, however, it will be in most cases.

I am interested in criteria on $\Omega$ to ensure that:

  1. The argument minimum defining $g$ is Lebesgue almost everwhere unique, such that $g$ is Lebesgue almost surely well-defined.

  2. $g$ is Lebesgue almost everywhere differentiable.

This problem is relevant in the context of for example shape restricted regression, see the article Meyer & Woodroofe (2000), where $\Omega$ is taken to be convex. In the case where $\Omega$ is convex, it is known that $g$ is well-defined, it is then ordinarily known as the "projection onto a convex set", also, $g$ is in this case a contraction and is in particular Lebesgue almost everywhere differentiable by Rademacher's theorem.

As regards the nonconvex case, it is easy to see that $g$ in general will not be everywhere well-defined, nor even everywhere continuous. If, for example, $\Omega$ is $\mathbb{R}^n$ minus a cone, moving $y$ along the axis of a cone, small perturbations of $y$ will make the projection onto $\Omega$ move wildly, and so $g$ will not have any chance of even being continuous on this axis.

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