Let $S$ be a closed bounded convex set with $S = \{ x : x_1^2 + x_2^2 + x_3^2 \leq 4, x_1^2 - 4x_2 \leq 0 \}$ and $y = (1,0,2)^T$.
I am trying to find the minimal distance from $S$ to $y$. The Closest-Point Theorem tells me there exists a unique $\bar{x} \in S$ such that $\bar{x}$ has minimal distance from $y$ and $(y - \bar{x})T(x - \bar{x}) \leq 0$ for all $x \in S$. So far I've determined a point in $S$ but not the minimum and can only determine an idea of what this minimal distance is. Assuming the problem involves just the 1st inequality constraint $x_1^2 + x_2^2 + x_3^2 \leq 4$ in the definition of $S$, I found the closest point $x_0$ on the outer region of the sphere with radius 2 and center at the origin to be
$x_0 = \frac{R}{||y^T||}y^T = \frac{2}{\sqrt{5}}(1,0,2) = (\frac{2}{\sqrt{5}}, 0, \frac{4}{\sqrt{5}})$
Calculating the distance from $y$ to this point gives:
$d_{unmodified} = (x_1 - 1)^2 + x_2^2 +(x_3 - 2)^2 = (1 - \frac{2}{\sqrt{5}})^2 + 0 + (2 - \frac{4}{\sqrt{5}})^2 \approx .055728$
Recall that this point is not in $S$ since it fails to meet the 2nd inequality constraint $x_1^2 - 4x_2 \leq 0$. So in attempting to get a better understanding of what distance a point $x$ in $S$ close to $x_0$ would have. By adjusting the value of $x_2$ and $x_3$ while keeping $x_1$ the same to satisfy both inequality constraints and then comparing this distance with the distance from $y$ to the outer point $x$ of the sphere formed by the 1st constraint would be the approach. Assuming this new point meets the equality constraint while keeping $x_1$ the same as in the unmodified version, $x_1^2 = \frac{2}{\sqrt{5}}$ and $x_1^2 - 4x_2 = 0$, we get $x_2 = 0.2$ and solving for $x_3$ from the 1st inequality gives approximately $x_3 \approx 1.77764$. So the new modified point in $S$ and close to the unmodified is then $(\frac{2}{\sqrt{5}}, 0.2, 1.77764)$ and has distance from $y$:
$d_{modified} = (x_1 - 1)^2 + x_2^2 +(x_3 - 2)^2 = (1 - \frac{2}{\sqrt{5}})^2 +0.2^2 + (2 - 1.77764)^2 = 0.10059 $.
To summarize the progress I've made so far, it would seem that the unique point with minimal distance from $S$ exists by the Closest Point Theorem and seems to have minimal distance $d_{min}$ somewhere inside the range of distances I've determined so far, i.e. $d_{min} \in (.055728,0.10059)$. I want to find the actual minimizing point but do not know how to proceed, any advice on possible computer or analytical methods to find this optimal minimal distance would be much appreciated.
Starting from Rob Pratt's answer, using Lagrange multipliers, the minimum value corresponds to the smallest real root (there are only $2$) of equation $$t^6-54t^5+976t^4-5976t^3+16629t^2-275994t+25826=0$$
Concerning the variables $x_1$ corresponds to the postive root of $$t^6+17t^4+80t^2-64=0$$ and this is nice since it is a cubic equation in $t^2$; that is to say that $$x_1=\sqrt{ \frac{14}{3} \cosh \left(\frac{1}{3} \cosh ^{-1}\left(\frac{2071}{343}\right)\right)-\frac{17}{3}}\qquad x_2=\frac 14 x_1^2\qquad x_3=\sqrt{4-x_1^2-x_2^2}$$ So, the minimum value is given by $$9-2x_1-\sqrt{64-x_1^2 \left(x_1^2+16\right)}$$