I am trying to show the following two statements are true:

(1) For any nonempty set $\Omega\subset\mathbb{R}^n$, the set $B$ consisting of points $y\notin\Omega$ where there is not a unique closest point $x\in\partial\Omega$ for each $y$ has $\mathcal{H}^n$-measure 0.

(2) This implies that the number of points $a\in A\subset \mathbb{R}^2$ such that $\mathcal{H}^{1}(\vec{n}(a)\cap B)>0$ is countable, where $\vec{n}(a)$ is the normal to $a\in A$ and $A$ is $C^1$ (i.e. it is locally the graph of a $C^1$ function from $\mathbb{R}$ to $\mathbb{R}$. More specifically, $A$ is a $C^1$ closed embedded 1-d submanifold in $\mathbb{R}^2$).

For (1), I would like help proving only this related fact: if the distance function $f$ to $\Omega$ is differentiable at a point $y\notin\Omega$, then there exists a unique closest point $x\in\partial\Omega$ to $y$.

I thought that I could get at this by saying that the graph of $f$ has a sharp corner wherever there is more than one closest point (and hence is not differentiable there). But, I am told this is not always true.

For (2), I thought that, since we're in $\mathbb{R}^2$, if we have an uncountable number of segments with positive $\mathcal{H}^{1}$ measure, then $B$ would have to have positive $\mathcal{H}^{2}$ measure since $B$ contains these segments. I'm told this is not true either. Please assist.

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