Does almost every point in Euclidean space have unique projection on any given set?

I am stuck at one point in my research, where I need to prove something which appears trivial to me, but cannot find a rigorous proof. I describe it below. Whenever I will say projection, I will mean the $$L^2$$ projection in Euclidean spaces.

We all know that the projection of a point on a closed, convex subset of the Euclidean space is unique. Now, if a set $$A \subseteq \mathbb{R}^n$$ is closed, we can easily show that the projection of a point $$p \in \mathbb{R}^n$$ on $$A$$ exists, but can find counterexamples, such that it is not unique. So, here is my first question (assume henceforth, that $$A$$ is a closed set):

1. Is it true that the set of all points $$p \in \mathbb{R}^n$$ that have more than one projection on the set $$A$$ (let us call that set $$\mathcal{P}(A)$$), has Lebesgue measure $$0$$?

It seems too much for question 1 to have an affirmitive answer for any arbitrary closed set $$A$$ (I do not even know whether the set $$\mathcal{P}(A)$$ is measurable, but if not, I can work with outer measures). So, here comes a simpler question.

1. If $$A$$ is the union of two convex sets, is it true that $$\mathcal{P}(A)$$ has Lebesgue measure $$0$$?

Even if this seems too much, I would really be happy to have an affirmitive answer (with a proof) to the following even simpler question:

1. If $$A$$ is the union of two polyhedra (a polyhedron is a finite intersection of half-spaces, and hence, is convex), does $$\mathcal{P}(A)$$ have Lebesgue measure $$0$$?

I intuitively feel that 3 must be correct, the reason being as follows. If a point $$p$$ has two distinct projections in $$A$$, then these projections must lie on two different faces of the union of the polyhedra, must be projections of $$p$$ on the respective faces too, and must be equidistant from $$p$$. The set of such point seems to be a lower dimensional hyperplane, which has Lebesgue measure $$0$$. But I cannot make this further rigorous. The problem is, I cannot seek help from linear algebra, as these faces are flats, and not even subspaces.

Any help (at least with answering question 3) will be greatly appreciated!