Hello I am trying to find an equation for a separating hyperplane to the set $S$ from an outer point $y$ defined as:
$S=\{x: x_{1}^{2}+x_{2}^{2}+x_{3}^{2}\le4, x_{1}^{2}-4x_{2}\le0\}$ and $y=(1,0,2)^{T}$
The set $S$ is just the region inside the sphere centered at the origin of radius 2 and inside the boundary set for $x_1$ and $x_2$ for the equation given by the inequality $x_{1}^{2}-4x_{2} \leq0$, the point $y$ is clearly not in this set since the value of $y = (1,0,2)$ is greater (so outside) of the inequality constraint $x_{1}^{2}+x_{2}^{2}+x_{3}^{2}\leq4$. I'm trying to find an equation of a hyperplane that would be between the sphere below and the point I have indicated.
So how would I go about setting up the problem I have in a formal manner and then solving for this separating hyperplane? Any help would be appreciated, thanks.
In addition how do I solve for the minimum separating distance from $y$ to $S$? Any insight on this would be appreciated as well.
In terms of finding a hyperplane separating the point from the sphere, I would start with finding a normal vector and point which can define the hyperplane. I think the easiest method would be to calculate the vector connecting the center of the sphere to the point in question; which in this case is just $y^T$. This vector is an outward normal of the sphere at the point where the line from the center of the sphere to $y^T$ intersects the sphere.
The intersection point $a^T$ can be calculated as:
$$ a^T = R\frac{y^T}{\left\lVert y^T \right\rVert} $$
Where R is the radius of the sphere.
A point 'in between' the sphere and $y^T$ can be calculated as:
$$ b^T = a^Tt + (1-t)(y^T-a^T) $$ $$ 0 < t < 1 $$ Where t is a scalar.
With respect to the second part of the problem, finding the minimum distance between $y^T$ and the set $S$, I would first check to see if the intersection point $a^T$ is in $S$. If so then then the minimum distance is just the distance between the point and the sphere. If not, then you will need to calculate the distance between the point $y^T$ and the 'extruded' parabola defined by $x_1^2 - 4x_2 <=0$.