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How do I compute the projection/closest vector to a subset? I have been thinking about this for far too long without any progress. If it helps, I am working in $Bbb{R}^2$, but I would like formalue in terms of norms and inner products, if possible.

For context, I am trying to prove that the figure eight is a deformation retract of the doubly punctured plane. And it is annoying that everything hinges on this annoyingly simple question. I have already shown that $overline{B}(0,1) setminus {p,q}$ is a deformation retraction of $Bbb{R}^2 setminus {p,q}$, where $p = (-frac{1}{2},0)$ and $q = (frac{1}{2},0)$. Now I just need to show that $overline{B}(0,1) setminus {p,q}$ deformation retracts to the union of the two discs with one centered at $p$, the other centered at $q$, but I am currently facing the obstacle discussed above.

EDIT: It just occurred to me that deformation retraction I had in mind won't be well-defined. Any point on the y-axis contained in $overline{B}(0,1)$ won't have a unique projection/closest vector in the union of the two open discs contained in $overline{B}(0,1)$...Hmm..need to rethink my approach...Of course, I wouldn't be opposed to any suggestions!

I do not really understand what you mean by "the projection/closest vector to a subset".

However, I shall explain how to get the desired strong deformation retraction. Let $p_{pm1}$ denote the points $(pm1,0) in mathbb{R}^2$, let the figure eight be the space $E = S_{+1} cup S_{-1}$, where $S_{pm1}$ is the circle around $p_{pm1}$ with radius $1$ and let the doubly punctured plane be the space $P = mathbb{R}^2 setminus { p_{+1}, p_{-1} }$. Define $r : P to E$ as follows. For $p = (x,y)$ set
$$r(p) =
begin{cases}
p_{+1} + dfrac{p - p_{+1}}{lVert p - p_{+1} rVert} & lVert p - p_{+1} rVert le 1 \
p_{-1} + dfrac{p - p_{-1}}{lVert p - p_{-1} rVert} & lVert p - p_{-1} rVert le 1 \
dfrac{2xp}{lVert p rVert^2} & lVert p - p_{+1} rVert ge 1, p ne 0, x ge 0 \
-dfrac{2xp}{lVert p rVert^2} & lVert p - p_{-1} rVert ge 1, p ne 0, x le 0
end{cases}
$$
Here $lVert - rVert$ denotes the Euclidean norm $lVert (x,y) rVert = sqrt{x^2+ y^2}$. Note that the denominator $lVert p - p_{pm 1} rVert$ does not vanish on $P$.

What happen geometrically? Let $D_{pm 1}$ = closed unit disk with center $p_{pm 1}$, $H_{pm 1}$ = right/left half plane minus the interior of $D_{pm 1}$.

The first two lines describe the radial strong deformation retractions from $B_{pm 1} = D_{pm 1} setminus { p_{pm 1} }$ to $S_{pm 1}$. In fact, for $p in B_{pm 1}$ we have $lVert r(p) - p_{pm 1} rVert = lVert dfrac{p - p_{pm 1}}{lVert p - p_{pm 1} rVert} rVert = 1$, and for $p in S_{pm 1}$ we have $p_{pm 1} + dfrac{p - p_{pm 1}}{lVert p - p_{pm 1} rVert} = p$.

Note that $B_{+1} cap B_{-1} = { 0 }$. Both line 1 and line 2 yield $r(0) = 0$.

The last two lines (together with $r(0) = 0$) describe strong deformation retractions of $H_{pm 1}$ to $S_{pm 1}$. This is done by shifting each point $p ne 0$ along the line through $0$ and $p$ until it reaches $S_{pm 1}$. To be formal, this line is given by $l_p(t) = t p$, and for $p in H_{pm 1} setminus { 0 }$ we must find $t$ such that $lVert t p - p_{pm 1} rVert = 1$. Easy computations show $t = pm dfrac{2x}{lVert p rVert^2}$, and in fact we defined $r(p) = l_p(t) = t p$.

Note that for $p in Y = H_{+1} cap H_{-1}$ = $y$-axis = set of points with $x = 0$ we have $r(p) = 0$.

Thus all four lines give us a consistent definition on the whole space $P$. It remains to show that $r mid_{H_{pm 1}}$ is continuous in $p = 0$. We have $lVert p - p_{pm 1} rVert ge 1$, i.e. $(x mp 1)^2 + y^2 ge 1$. This is equivalent to $pm 2x le x^2 + y^2$ which means $2lvert x rvert le lVert p rVert^2$ since $p in H_{pm 1}$. Hence $lVert r(p) rVert = dfrac{2 lvert x rvert}{lVert p rVert} le lVert p rVert$ for $p ne 0$. This immediately implies continuity.

We now have constructed a retraction $r$. To see that it is a strong deformation retraction, define a homotopy
$$H : P times I to P, H(p,t) = (1-t)p + tr(p) .$$
It is readily verified that in fact $H(p,t) ne p_{pm 1}$ for all $(p,t)$ (check all 4 lines in the definition of $r$). This is a homotopy from $id_P$ to $r$ which is stationary on $E$.