Please see the image for best illustration of the task.
I have two hyperrectangles, $\text{R1}$ and $\text{R2}$, whose exact location and size is arbitrary. Now, my task is to construct a bounding circle, something like $\text{C4}$, such that its area includes any other circle formed by any of the edges of the rectangles (like $\text{C2}$ or $\text{C3}$).
My initial attempt was to construct a bounding rectangle that includes $\text{R1}$ and $\text{R2}$ and draw a circle $\text{C1}$ with the radius set to half the width of the longest distance between its edges. But as the drawing shows, that would ignore parts of the areas of $\text{C2}$ and $\text{C3}$.
Is there a mathematical way to determine a minimal radius value for the bounding circle $\text{C4}$ that would cover all the inner circles. Also, I am working in the multidimensional space, so the solution should not restrict itself to $2$ dimensions.
Thanks :)
I am not very knowledgeable in this subject, but it appears that a general version of this question is a hard problem.
Note that all of the points of a hyperrectangle will be inside the hypersphere iff each of their vertices are inside the hypersphere. Using we can convert our problem into a standard bounding sphere problem with $2\times 2^d$ points where $d$ is the dimension of the hyperrectangles. It appears that there are several algorithms for this problem of varying degrees of expected closeness to optimality. It's possible that there is a better method given that you have hyperrectangles, but I couldn't find any.
Note that you can at least get trivial bounds on the size of the sphere - the upper bound given by the method you describe to find $\text{C1}$, and the lower bound given by a sphere having a radius of half the largest distance between the $2\times2^d$ points.