# Separating hyperplane from a set to an outer point

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 $$x_1^2 - 4x_2 <=0$$.